INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Hormones, neurotransmitters, and growth factors that activate phospholipase C generate a bifurcating signal with Ins(1,4,5)P₃ at one arm and diacylglycerol (DAG) at the other (Rhee, 2001). Although DAG activates protein kinase C (Nishizuka, 1988), InsP₃ mobilizes calcium via the endoplasmic reticulum based InsP₃ receptor (Berridge, 1993). Intracellular levels of these secondary messengers depend on a balance between their rate of formation and rate of removal, which channels them back to lipid resynthesis. Ins(1,4,5)P₃ is metabolized in cells by an interplay of phosphatases and kinases (Shears, 1989) that result in production of inositol phosphates ranging from inositol monophosphates to inositol heptaphosphates and octaphosphates (Irvine and Schell, 2001). Understanding the kinetics of this complex metabolic network for InsP₃ is fundamental to deciphering its role in shaping intracellular calcium dynamics.

InsP₃ functions as an important secondary messenger (Berridge and Irvine, 1989; Streb et al., 1983) that binds to InsP₃ receptors embedded in the endoplasmic reticular membrane and mediates calcium release into the cytosol (Taylor and Richardson, 1991). This release can elevate calcium from 10 to 100 nM basal levels to several micromolar stimulated levels (Bootman and Berridge, 1995). Depending on the cell type, the calcium waveform can either exhibit a single peak or can oscillate with multiple spikes (Berridge and Irvine, 1989; Tsien and Tsien, 1990; Meyer and Stryer, 1991). Isolated calcium puffs generated locally in the cytosol can propagate throughout the cell as waves (Lechleiter et al., 1991; Parker and Yao, 1991) and activate various cell physiological processes (Berridge, 1993) including differentiation, proliferation, vesicle release, and sensory perception.

Like InsP₃, its metabolic products also play essential roles in cellular function, although they have not been as rigorously assessed as InsP₃. For instance, evidence has been accumulating for the role of Ins(1,3,4,5)P₄ in facilitating InsP₃-mediated Ca²⁺ release (Morris et al., 1987; Cullen et al., 1990; Smith et al., 2000). It was recently shown that calcium release activated calcium current (I_CRAC) (Hoth and Penner, 1992) is enhanced due to the inhibitory effect of Ins(1,3,4,5)P₄ on InsP₃ 5-phosphatase (Hermosura et al., 2000). The Ras signaling pathway is also affected by Ins(1,3,4,5)P₄ function. GTPase activating proteins, GAP1m and GAP1IP₄BP, gain activity by specifically binding InsP₄ (Cullen et al., 1995; Fukuda and Mikoshiba, 1996).

Ins(3,4,5,6)P₄ also functions as a secondary messenger that regulates calcium activated chloride efflux. It inhibits a plasma membrane chloride channel (Vajanaphanich et al., 1994; Shears, 1998) and is thus involved in osmoregulation. Further, its cellular levels are modulated by an InsP₃ isoform Ins(1,3,4)P₃, which is generated by the degradation of Ins(1,3,4,5)P₄ (Yang et al., 1999).

InsP₅ and InsP₆ (phytic acid) belonging to the inositol high polyphosphate series (IHPS), play essential regulatory roles in endocytosis and exocytosis. In vitro, they have been shown to interact in varying affinities with assembly proteins important in clathrin-mediated endocytosis and with synaptotagmin domains involved in synaptic vesicle trafficking (Fukuda and Mikoshiba, 1997). "High energy" pyrophosphates are equally important for intracellular trafficking. PP-InsP₅ is the most potent known inhibitor of AP-180-mediated clathrin cage assembly (Ye et al., 1995). There is also evidence that InsP₅ and InsP₆ may function as extracellular signals to regulate blood pressure and heart rate (Vallejo et al., 1987).

In addition to these specific signaling actions, inositol phosphates have also been shown to bind crucial cellular proteins like the cytoskeletal element vinculin, the signaling molecule Bruton's tyrosine kinase (Btk), and the cell adhesion molecule myelin proteolipid protein (Fukuda and Mikoshiba, 1997). Active research is being conducted to find the exact physiological significance of these varied binding properties of inositol phosphates.

In parallel with research on the functional relevance of inositol phosphates, several key enzymes in the metabolic cascade of InsP₃ have been identified (Majerus, 1992). These enzymes regulate the cellular concentrations of inositol phosphates under basal and stimulated conditions. They are all kinases and phosphatases that sequentially add or remove phosphate groups from the inositol ring, exhibiting high specificity for their substrates. They show extensive cross-talk among themselves and are subjected to regulation by inositol phosphates in the metabolic network that are not their immediate substrates and products. Some of these enzymes are also regulated by general signaling molecules such as calcium, protein kinase C (Sim et al., 1990), calmodulin (CaM), and calcium-calmodulin activated protein kinase type II (CaMKII) (Communi et al., 1997). Another dimension to interactions among inositol phosphate metabolism enzymes is their varied spatial distribution (Soriano et al., 1997). Most of them are present in the cytosol but some are attached to the plasma membrane whereas others such as multiple inositol polyphosphate phosphatase are compartmentalized in the ER (Chi et al., 2000; Nogimori et al., 1991). This distribution alters accessibility toward substrates and regulatory molecules. Further complexities arise because of the presence of multiple isoforms of the same enzymes, whose expression levels differ from one cell type to another. Although most of these kinases and phosphatases have strict substrate specificity, some also catalyze multisubstrate reactions.

We have constructed a biochemical model for the cellular metabolism of Ins(1,4,5)P₃. To generate the mass action model, we have made use of published biochemical data (see Supplementary Information) on enzyme purification and characterization, primarily from brain tissue studies. For understanding how metabolism modulates the levels of the various inositol phosphates under basal or stimulated conditions, the inositol phosphate network has been integrated with an existing model for Ins(1,4,5)P₃ generation via phospholipase C activation (Bhalla and Iyengar, 1999). Stimulation is provided to the system as an external square pulse of glutamate transduced via the metabotropic glutamate receptor.

InsP₃ releases calcium from ER stores, and calcium can feed back onto both the production and degradation of InsP₃ (Harootunian et al., 1991; Communi et al., 1997). To account for this feedback, we incorporated a simplified model for the InsP₃ receptor and for calcium homeostasis. We found that the simple InsP₃ receptor model with a single InsP₃ binding step and no calcium feedback onto the receptor can only generate a nonoscillatory calcium response. Hence, to study interactions between InsP₃ metabolism and oscillatory calcium dynamics we adapted an existing model for detailed InsP₃ receptor kinetics developed by Othmer and Tang (Tang et al., 1996). The Othmer-Tang model incorporates both positive and negative Ca²⁺ feedback onto the InsP₃ receptor and produces periodic calcium spikes upon stimulation of InsP₃ levels.

Simulations of our InsP₃ metabolism model allow us to gauge how various inositol phosphates respond to a G-protein coupled receptor (GPCR) stimulus as a function of both concentration and time. We find that the response of InsP₃ is modified both by the presence of its metabolic cascade and by oscillations of calcium. Ins(1,3,4,5)P₄ emerges as a prominent regulatory molecule both for InsP₃ dynamics and calcium release.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

A block diagram representation of the InsP₃ metabolism model is presented in Fig. 1 A. Stimulus to the model was provided via glutamate (Glu) at the metabotropic glutamate receptor (mGluR). This lead to binding of GTP to the G subunit of G-protein. The activated G-protein then activated phospholipase C (PLC) , which generated InsP₃ and DAG from phosphatidylinositol bisphosphate (PIP₂). Whereas DAG activated PKC, stimulated levels of InsP₃ acted on the ER InsP₃ receptor and released stored Ca²⁺. Calcium, in turn, activated CaM, CaMKII, and PKC, which regulated Ins(1,3,4,5)P₄ formation among the higher phosphates. Ca²⁺ also activated PLC  by positive feedback and modulated enzymes within the lower inositol phosphate cascade. Termination of the GPCR signal was accelerated by the GTPase-activating protein (GAP) activity of PLC .

View larger version (29K): [in this window] [in a new window]   FIGURE 1   (A) Block diagram of complete model. The glutamate stimulus is provided at the metabotropic glutamate receptor (mGluR). G-protein activated PLC  generates InsP3 and DAG from PIP2. The GAP activity of PLC  accelerates termination of the G-protein signal. Whereas DAG activates PKC, InsP3 acts on the ER InsP3 receptor and releases Ca2+ from stores. Stimulated levels of cytosolic calcium activate Calmodulin, CaMKinase II, and PKC, which regulate Ins(1,3,4,5)P4 formation among the higher phosphates. Ca2+ also has positive feedback onto PLC  and modulates enzymes within the lower phosphate cascade. , Activation; inhibition. (B) Diagram representing modeled details of InsP3 metabolism. Pointer symbols used are as follows: , Michaelis-Menten enzymes; , reversible enzymes; <, enzymes that have Michaelis-Menten kinetics in the basal form but are reversible in the activated form; >, simple reactions; , reversible simple reactions; , competitive enzyme inhibition; ---, a composite process where substrate is transported to ER, reaction is catalyzed by ER based enzyme, and product is transported back to the cytosol. Numbers next to symbols depict enzyme/reaction names. Identical numbering on different enzyme reactions indicates that the same enzyme catalyzes many unique reactions with distinct rates for each conversion. Multiple pointer symbols from one inositol phosphate to another represent the different ways in which the same interconversion is achieved. The box on the left depicts Ins(1,4,5)P3 3-kinase and covalent modifications to it by PKC, CaM, and CaMKII, that modify enzyme activity as shown. 1, Ins(1,4,5)P3 3-kinase; 2, InsP 5-phosphatase1; 3, Ins(1,4,5)P3 5-phosphatase2; 4, Ins(1,3,4,5)P4 3-phosphatase; 5, InsP 1-phosphatase; 6, InsP1 phosphatase; 7, InsP 4-phosphatase; 8, Ins(1,3)P2 3-phosphatase1; 9, Ins(1,3)P2 3-phosphatase2; 10, Ins(3)P1 phosphatase; 11, multiple inositol polyphosphate phosphatase; 12, Ins(1,3,4)P3 5,6-kinase/Ins(3,4,5,6)P4 1-kinase/InsP5 1-phosphatase; 13, Ins(1,4,5,6)P4 6-phosphatase; 14, Ins(1,3,4,6)P4 5-kinase, 15, Ins(1,4,5,6)P4 3-kinase; 16, InsP5 1-phosphatase; 17, InsP5 3-phosphatase; 18, InsP5 2-kinase/InsP6 2-phosphatase; 19, InsP6 kinase1; 20, InsP6 kinase2; 21, PP-InsP5 kinase; 22, Diphosphoinositol polyphosphate phosphohydrolase; 23, PP-InsP4 phosphatase.

Fig. 1 B illustrates details of inositol phosphate metabolism that were incorporated in our model. The metabolism of Ins(1,4,5)P₃ was modeled as a network of enzymes, including details of the regulation of these enzymes. As shown in Fig. 1 B, there were numerous instances of enzyme regulation by competitive inhibition from different inositol phosphates. Detailed regulation of InsP₃ 3-kinase via CaM binding and phosphorylation by PKC and CaMKII was also part of the model. Interactions among members of the network model were represented as chemical reactions that were either simple reactions characterized by a K_d (dissociation constant) or a K_eq (equilibrium constant) (Eq. 1) or were enzymatic reactions.

Simple reactions were of the form:

(1)

In this formulation, interactions are completely specified by the initial concentrations of the reactants (A and B) and the product (C), by the rate constant for the forward reaction, k_f, and the rate constant for the back reaction, k_b.  represented the time course of the simple reaction. Its exact value, which can be influenced by various factors in a reaction network, was initially approximated during model building and subsequently validated through simulations.

Eq. 1 can also be represented as a differential equation given by: d[A]/dt = k_f [A][B] + k_b[C] and similar equations for the other molecules that participate in the reaction.

Enzyme reactions modeled in the network mostly followed Michaelis-Menten kinetics (Eq. 2), but some enzymes were modeled with reversible kinetics.

The Michaelis-Menten scheme for enzyme-catalyzed reactions was of the form:

(2)

This standard formulation for enzymes is a special case of two reactions in sequence with the assumption that the final step is irreversible. It is specified by three rate constants k₁, k₂, and k₃, and the initial concentrations of the enzyme E, substrate S, and product P. Unless actual rate constants were provided in literature, k₂ was taken as four times k₃. Most enzyme models are insensitive to the exact value of this ratio (Bhalla, 1998). We validated this in our model also by testing k₂:k₃ ratios ranging from 0.4 to 40 without significant variation in simulation results (data not shown).

Reversible enzyme reactions were of the form:

(3)

Here K and K represent the distinct affinities of the enzyme for the substrate and product respectively, and V and V represent the maximal velocities of the forward and backward reactions. Enzymes were modeled as reversible reactions when the back flux from the reaction products to the substrates was over 5% of the forward flux. The value of the fourth rate constant k₄, unless available in literature, was derived by free energy calculations. For a reaction at equilibrium, the standard free energy change G° (kJ/mol) is related to K_eq (equilibrium constant for the reaction) by:

(4)

Further K_eq is related to the rate constants of the reversible enzyme reaction by:

(5)

We extracted k₄ for each enzyme reaction for G° values ranging from 10 kJ/mol to 25 kJ/mol. This G° range was approximated from values for glucose phosphorylation (Jencks, 1976) as the G° for phosphorylation of the inositol ring has not, to our knowledge, been reported in the literature. We chose a wide range of G° values to span the likely range of k₄ rates. Glucose was chosen due to its structural similarity to inositol. This similarity is quite close, and in some cases glucose serves as a cellular precursor to inositol phosphates (Irvine and Schell, 2001).

In this way all enzyme reactions depicted in Fig. 1 B by symbols were modified from the Michaelis-Menten scheme to the reversible kinetics scheme. In general the reversibility of reactions did not introduce large changes in the model kinetics, but there are some interesting exceptions that are discussed in the Results section.

Parameters for all enzymes modeled were derived from published biochemical experiments that involve protein purification and characterization of enzyme kinetics. Michaelis-Menten enzymes were characterized by their K_m and V_max, and their initial concentrations were obtained from biochemical purification series and quantitative immunoprecipitation. In most cases, measurements across different published reports in literature allowed us to cross check the values of the constants we used. For simple reactions, k_f and k_b were constrained by time courses and dose response curves. Certain reactions within the inositol phosphate metabolic network that have been described in literature as enzymatic reactions but for which enzyme kinetics have not been characterized in mammalian systems had to be modeled as simple reactions. Rates for such reactions were constrained by equilibrium concentrations of their reactants and products. The incorporation of these simple reactions into the model introduces parameters that have not been tested experimentally. Nevertheless, the equilibria for these reactions are tightly constrained by the known steady-state levels of reactants in the InsP₃ metabolism cascade. Without these reactions, depicted in Fig. 1 B by numbers 10, 13, 14, 16, 17, 18, and 23, components of the metabolic network do not equilibrate in simulation runs due to unbalanced source-sink relationships among them. We acknowledge that precise kinetics of these reactions require validation through experiments.

Detailed parameters for all modeled reactions are presented in the Supplementary Information and on the DOQCS website (http://doqcs.ncbs.res.in, accession 31-32). A sample parameter set that describes enzyme 2 and its inhibition by InsP₅ in Fig. 1 B is shown in Table 1. Table 2 enumerates the total number of molecules, reactions, Michaelis-Menten enzyme activities, and channels present in our model. The non-Osc-model in Table 2 refers to a version of the network model wherein details of InsP₃ receptor kinetics were not incorporated and which showed nonoscillatory dynamics for InsP₃-mediated Ca²⁺ release. The Osc-model refers to the network model that included detailed InsP₃ receptor kinetics. In this model, kinetic parameters for the InsP₃ receptor were closely based on the Othmer-Tang calcium dynamics model (Tang et al., 1996). The Osc-model also differed from the non-Osc-model in basal Ca²⁺ levels and interactions between the ER sequestered and extracellular calcium pools. These parameter modifications were made with the sole objective to generate cytosolic Ca²⁺ oscillations and have been enumerated in the Supplementary Information.

Input to the signaling network was delivered as square pulses of glutamate at the mGluR. The model used for representing mGluR does not include receptor inactivation, desensitization, or receptor recycling details. It is an approximation of actual receptor kinetics intended to serve as a stimulus generator for the main focus of our model, InsP₃ metabolism. In our model, stimuli of amplitude  0.5 µM glutamate produced saturating responses. The system was allowed to settle to a steady state before being stimulated, and input delivery was followed by a poststimulatory run that ensured restoration of steady state.

A sensitivity analysis was performed to investigate how sensitive the model was to the numerous parameters used to build it. For this analysis, parameters were classified in five categories of initial concentration, reaction dissociation constant (K_d), reaction time course (), enzyme Michaelis constant (K_m), and enzyme maximal velocity (V_max). Values of all parameters in each category were systematically scaled 0.1, 0.2, 0.5, 0.667, 0.833, 1.2, 1.5, 2, 5, and 10 times their original value and the model subjected to simulation runs. Variability in results for this 100-fold range (0.1-10) of all parameter values were analyzed by generating scatter plots for logarithmic fold changes in outputs for calcium, Ins(1,4,5)P₃, Ins(1,3,4,5)P₄, and Ins(1,3,4)P₃. For the model that included detailed InsP₃ receptor kinetics, the scatter plot readout was the frequency of calcium spikes generated within stimulus time. For ease of analysis, all scatter plots were normalized with respect to the control output. The control refers to the basic scheme of the model with the original parameter values.

Two global sensitivity analyses were also performed to investigate the effects of alterations in a combination of parameters on the model behavior. The first global analysis focused on effects of temperature alteration on the system. This was based on the premise that an ~10°C rise in temperature accelerated reaction rates by twofold. Simulations were performed to assess the effects of 10°C rise and 10°C fall in temperature on all reaction rates. The second global sensitivity analysis targeted all energy consuming reactions, i.e., kinases and ATPases, which would be affected by ATP depletion in the system. Because for most reactions ATP was not explicitly included as a reactant species, a change in ATP concentration to x times its original value was approximated by scaling enzyme K_m by a factor of 1/x, and by changing the k_f and k_b of ATP dependent reactions by a factor of x and 1/x, respectively. Simulation runs were performed for ATP depletion to 0.5 and 0.15 times its original concentration, which represent physiological drops in cellular ATP levels (Kahlert and Reiser, 2000). Sensitivity analysis outputs were monitored for Ca²⁺, Ins(1,4,5)P₃, Ins(1,3,4,5)P₄, and Ins(1,3,4)P₃ and are discussed at the end of the Results section.

All numerical computations were performed using a graphical interface-Kinetikit (version 7) on the General Neural Simulation System-GENESIS (Bhalla, 1998). Computations were carried out on PCs running Linux. The exponential Euler formulation was used for integration (MacGregor, 1987). Numerical accuracy of the computations was verified by comparing the results for simulations that had been run at different time steps. The model provided convergent solutions for the range of time steps used in the study. For analysis of the model output, a time step of 0.2 ms was used. At initiation of the simulation run and at all transient points wherein steady state of the model was perturbed by a stimulus input, a fine time step of 10 µs for 10 s was used. The temporal characteristics of some of the output curves are described in terms of rise time of the response to stimulus, response latency, and response width. We define rise time of a response to stimulus as the time taken to progress from 20% to 80% of the maximal response height. Response latency is defined by the time taken to achieve 20% of maximal response height from stimulus onset, and response width represents the full width of the response at half maximal response height.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Responses of components of the Ins(1,4,5)P₃ metabolic pathway

We first characterized the effects of a PLC -activating stimulus on the components of the Ins(1,4,5)P₃ metabolism model. For this purpose, the non-Osc-model was initially allowed to achieve steady state by running the model without stimulation for a period of 1000 s. Stimulus was then provided as a square pulse of 0.5 µM glutamate for 30 s. Poststimulation model responses were simulated for 1000 s.

Components of the InsP₃ metabolic pathway showed varied responses to stimuli with respect to both amplitude and time of response. These are depicted in Fig. 2.

View larger version (31K): [in this window] [in a new window]   FIGURE 2   Responses of the various inositol phosphates in the non-Osc-model to a 30-s pulse of 0.5 µM glutamate. The concentration axis is scaled differently in each panel. Response latency (RL) and response width (RW) are calculated as described in the methods and reported for selected traces. (A) Responses of the two InsP3 isoforms are markedly different. Ins(1,4,5)P3 shows a fast transient response (RL, 22 s; RW, 20 s). Ins(1,3,4)P3 has a slow prolonged rectangular response (RL, 32 s; RW, 246 s). (B) Responses of InsP2s mimic responses of InsP3s from which they are derived. (C) Responses of InsP1s whose high basal levels are maintained by back flux from inositol. (D and E) Inositol high polyphosphates: Ins(1,3,4,5,6)P5, InsP6 and PP-InsP4, and inositol pyrophosphates: PP-InsP5 and bis-PP-InsP4 show negligible stimulus responses. (F) Ins(1,3,4,5)P4 shows the largest (60-fold) response to stimulus (RL, 52 s; RW, 139 s). (G) Responses of InsP4s: Ins(1,3,4,6)P4, Ins(1,4,5,6)P4, and Ins(3,4,5,6)P4, of which the Ins(3,4,5,6)P4 response is physiologically important for chloride channel regulation. (H) Biphasic Ca2+ response to the Ins(1,4,5)P3 generating stimulus.

Responses of Ins(1,4,5)P₃ and its lower phosphates

Responses of Ins(1,3,4)P₃ and its lower phosphates

Responses of Ins(1,3,4,5,6)P₅ and other inositol high polyphosphates

Responses of inositol tetrakisphosphates

Calcium response in the model and incorporation of oscillations

The primary function of Ins(1,4,5)P₃ as a secondary messenger is to release calcium from intracellular stores in response to external stimuli. We used a 0.5-µM glutamate stimulus lasting 30 s to monitor the calcium output. As seen in Fig. 2, the biphasic InsP₃ response to stimulus in the non-Osc-model resulted in biphasic calcium release. Such a response with an early peak phase followed by a lower plateau-like phase has been reported for calcium under physiological conditions (Lambert and Nahorski, 1990). Peak stimulated levels of Ca²⁺ reached ~0.9 µM and were ~12-fold above basal.

Various mechanisms have been hypothesized for the periodic oscillations of cytosolic calcium seen in different cell types (Berridge, 1990). Positive feedback of calcium onto PLC  that generates InsP₃ underlies the calcium oscillation mechanism shown in certain experiments (Harootunian et al., 1991) and models (Meyer and Stryer, 1988). Although our non-Osc-model incorporated such feedback, it failed to display Ca²⁺ oscillations. This probably resulted from different parameter representations across models and cell types. Hence, to study interactions between molecules in the inositol phosphate metabolic cascade and oscillatory calcium dynamics, we appended the Othmer-Tang calcium oscillation model (Tang et al., 1996) to our InsP₃ metabolism model. As mentioned previously in the Materials and Methods section, we refer to the resultant composite model as the Osc-model. The Othmer-Tang model is characterized by both a positive and a negative feedback from calcium onto its store release channel, the InsP₃ receptor. The channel in its conducting state has both calcium and InsP₃ bound. This mobilizes calcium from the ER to the cytosol. An excess calcium accumulation in the cytosol results in more calcium being bound to the receptor. This leads to channel inactivation.

The InsP₃ receptor response to increasing InsP₃ and calcium concentrations, in the nanomolar to micromolar range, was plotted as the normalized fraction of channels open versus the log of concentration (Fig. 3). The plots show a bell-shaped curve for InsP₃ receptor response to calcium, and a saturating curve reflects dependence of response on InsP₃ input. These are similar to the plots generated by Tang et al. (1996).

View larger version (28K): [in this window] [in a new window]   FIGURE 3   Responses of the InsP3 receptor (kinetics adapted from the Othmer-Tang model) as a function of (A) InsP3 concentration and (B) Ca2+ concentration. Ca2+ levels in µM used in A to generate the InsP3 function curves are shown on the graph. InsP3 levels used in B to generate the Ca2+ function curves (from the lowermost curve upward) are: 0.25, 0.5, 0.75, 1.0, 2.0, 5.0, 10, 20, 50, and 100 µM, as alternating dashed and solid lines, respectively. Concentration (µM) values represented on the x axis are on a logarithmic scale. Values on the y axis represent the steady-state concentration of active InsP3 channels normalized against the maximal concentration of active channels obtained with 0.1 µM Ca2+ in A and 100 µM InsP3 in B. The maximal percentage of channels open in A and B are 8% and 9.5%, respectively.

In our Osc-model basal calcium levels are ~20 nM, which is below the physiological range. Low basal calcium levels close to 0 nM are also used by Othmer and Tang for their model and are necessary for sustaining the oscillatory response. For maintaining calcium at such low levels we had to modify parameters for interactions between cytosolic and extracellular calcium, such as the store-operated calcium entry and the plasma membrane calcium pump (see Supplementary Information). These parameters may not be an accurate representation of experimental observations.

A comparison of the calcium responses within the non-Osc-model and the Osc-model to the 30-s stimulus pulse of 0.5 µM glutamate is provided in Fig. 4. The single calcium transient in the non-Osc-model is replaced by four calcium spikes in the Osc-model. These two models are used separately to study interactions between calcium and InsP₃ metabolism in further sections of this paper. Below we shall also analyze the differences that arise in these interactions due to the oscillatory nature of calcium.

View larger version (17K): [in this window] [in a new window]   FIGURE 4   Comparison of Ca2+ responses generated in the (A) non-Osc-model and (B) Osc-model to a 30-s 0.5 µM glutamate stimulus (solid bar).

Effects of glutamate stimulus intensity on InsP₃ buildup and calcium release

Cellular systems respond differently to different intensities of stimuli. The input parameters can modulate the output in terms of amplitude, time, and frequency of response. Here we analyze effects of different intensities of the glutamate stimulus on InsP₃-mediated calcium release. Fig. 5 A shows the calcium response within the non-Osc-model to 30-s pulses of glutamate stimuli of magnitude: 5 nM, 10 nM, 50 nM, 0.1 µM, and 0.5 µM. The time curves show that increasing stimulus concentrations affect both the amplitude and rise time of the calcium peak. These response parameters are plotted as a function of glutamate stimulus concentration in Fig. 5, B and C. The peak Ca²⁺ release increased nonlinearly with increasing stimulus intensity. At low glutamate concentrations, up to 10 nM, the Ca²⁺ change above basal was almost negligible. The magnitude of Ca²⁺ release increased rapidly in the 10- to 100-nM input range and then saturated at a peak ~1 µM Ca²⁺ at higher stimulus intensities. The inset of Fig. 5 B shows a plot for the peak InsP₃ response to the same stimulus protocol. The similarity between the stimulus function curves for peak Ca²⁺ and InsP₃ indicates that calcium release is a direct consequence of InsP₃ buildup. At higher concentrations of glutamate, there is higher occupancy of the mGluR, which leads to greater PLC  activation and buildup of more InsP₃. The plots show that there exists a relatively sharp stimulus threshold for InsP₃ generation and the subsequent Ca²⁺ release. In our model, this threshold is determined by the positive feedback of Ca²⁺ onto PLC . PLC  has maximal activity when activated by both Gq and Ca²⁺. In this form it catalyzes sufficient InsP₃ formation to elevate cytosolic Ca²⁺ levels over 10-fold above basal.

View larger version (14K): [in this window] [in a new window]   FIGURE 5   Ca2+ release as a function of intensity of glutamate stimulus. (A) Thirty-second pulses of 5 nM, 10 nM, 50 nM, 0.1 µM, and 0.5 µM glutamate are used as stimuli. The pulse amplitude in nanomolars is shown on the graph. For the non-Osc-model, rise time of the Ca2+ response decreases and amplitude increases with increasing stimulus intensity. (B) Peak Ca2+ responses (as obtained in A) are a nonlinear function of stimulus intensity. Inset shows a corresponding plot for the InsP3 response. Glutamate concentrations (µM) represented on the x axis have a logarithmic scale. (C) Positive feedback of Ca2+ onto PLC  mediates the decrease in rise time of the Ca2+ response. The solid and dashed lines plot the rise time of the Ca2+ response as a function of stimulus intensity, in presence and absence of the positive feedback of Ca2+ onto PLC , respectively. Stimulus protocol is same as that used for A. (D) Frequency of Ca2+responses in the Osc-model are a nonlinear function of stimulus intensity. Stimuli used are 2-min pulses of 1 nM, 5 nM, 10 nM, 50 nM, 0.1 µM, and 0.5 µM glutamate, represented on a logarithmic scale on the x axis.

Fig. 5 C shows that the rise time of the Ca²⁺ response decreases with increasing stimulus intensity. This temporal acceleration of the peak response also results from the positive feedback of Ca²⁺ onto PLC , which leads to regenerative calcium release. We tested this by removing the Ca²⁺ facilitation of PLC enzyme activity from our model. This resulted in an almost constant rise time of the Ca²⁺ response at any stimulus strength (Fig. 5 C).

We also performed simulations to study the effect of stimulus intensity on the Osc-model. Two-minute pulses of 1 nM, 5 nM, 10 nM, 50 nM, 0.1 µM, and 0.5 µM glutamate were used as stimuli. The frequency of calcium oscillations was taken as readout of the stimulus effect (Fig. 5 D) (Berridge, 1990). Like the calcium release in the non-Osc-model, the relationship between stimulus input and simulation output, oscillation frequency in this case, is nonlinear. A 1 nM glutamate stimulus produced no oscillations, whereas 0.1 and 0.5 µM stimuli produced a saturating response frequency of 2/min. The increase in frequency of oscillations is expected because increasing stimulus intensities produce increasing InsP₃ in the cytosol. At higher InsP₃ concentrations the bell-shaped conductance curve of the InsP₃ receptor with respect to calcium shows faster response kinetics than at lower concentrations (Fig. 3 B). This results in the greater number of calcium spikes within the same time period.

Effect of calcium oscillations on the InsP₃ metabolic cascade

It is known that the pattern of calcium oscillations regulates downstream events such as gene expression (Dolmetsch et al., 1998). Using our simulations we wanted to investigate whether oscillatory calcium dynamics also modify InsP₃ and other inositol phosphates by feedback. This is important as the different physiological functions of the various inositol phosphates could be temporally affected by the oscillations. For this purpose, the Osc-model was subjected to a stimulus pulse of 0.5 µM glutamate for 2 min. The calcium response generated by the pulse is represented in Fig. 6 A. Within stimulus time cytosolic calcium displayed oscillations of uniform amplitude and similar interspike interval. Three residual Ca²⁺ spikes were also observed after stimulus termination, before InsP₃ was restored to basal levels. These residual spikes emerged after a time lag that represented the recovery time for the depleted ER calcium stores after the stimulus trigger. The 0.5 µM glutamate stimulus had emptied the stores to a level that Ca²⁺ could not be released until ER levels were adequately restored by the Ca²⁺-ATPase pump.

View larger version (23K): [in this window] [in a new window]   FIGURE 6   Responses of inositol phosphates in the Osc-model. Stimulus is a 2-min pulse of 0.5 µM glutamate (solid bar). (A) Oscillatory response of Ca2+. Oscillations are of frequency ~2/min and persist beyond stimulus time. (B) InsP3 levels oscillate in synchrony with Ca2+. (C) Ins(1,4,5)P3 oscillations are transmitted to its dephosphorylated products: Ins(1,4)P2 and Ins(4)P1, which also oscillate periodically with Ca2+, but less prominently.

Fig. 6, B and C depict the corresponding responses of Ins(1,4,5)P₃, Ins(1,4)P₂, and Ins(4)P₁ to the 2-min 0.5 µM glutamate pulse that generates the calcium response shown in Fig. 6 A. We observed that oscillatory calcium, which fed back onto PLC -mediated InsP₃ generation, produced oscillations in InsP₃. We suggest that oscillations in cytosolic InsP₃ are possible, but instead of being a requirement for calcium release, they are generated as a result of feedback of calcium pulses. It has been shown experimentally that InsP₃ levels oscillate in cells (Hirose et al., 1999). At the same time, it has also been shown that nonoscillatory InsP₃ dynamics produce oscillations in calcium (Wakui et al., 1989). The Othmer-Tang model is also based on this formulation, wherein a square pulse of InsP₃ can generate pulsatile calcium release. Thus, our simulations, which combine the Othmer-Tang model with Ca²⁺ feedback onto InsP₃ generation, indicate that InsP₃ and calcium oscillations can coexist. However, it is not necessary that oscillations in calcium are brought about by oscillations in InsP₃.

Hirose et al. (1999) demonstrated that InsP₃ oscillates in a spatiotemporal manner in cells exhibiting calcium oscillations. These researchers showed a spatial component to the InsP₃ oscillations, wherein InsP₃ is translocated through the cytosol from its site of synthesis near the cell membrane to its site of action at the ER. They also discuss the role of a Ca²⁺-mediated negative feedback in the generation of InsP₃ oscillations. As we have not incorporated any component in our model to account for spatial diffusion of InsP₃, we cannot comment on its spatial dynamics. We have also not included any negative feedback of calcium onto InsP₃ synthesis in our model.

The biochemical connections in the InsP₃ metabolic network transmit InsP₃ oscillations to its dephosphorylated metabolites (Fig. 6 C). Like InsP₃, the oscillations in Ins(1,4)P₂ and Ins(4)P₁ were also synchronous with calcium. Apart from Ins(1,4,5)P₃, Ins(1,4)P₂, and Ins(4)P₁, we observed negligible or no oscillatory perturbations for other inositol phosphates (not shown). This is expected from the response time courses of inositol phosphates downstream of InsP₃ phosphorylation (Fig. 2), which were so slow that oscillatory effects, if any, averaged out.

Hence, from our simulations we infer that the InsP₃ oscillations that we observe to be synchronous with calcium are a result of the competition between InsP₃ degradation and positive calcium feedback on InsP₃ synthesis.

Modification of Ins(1,4,5)P₃ response and calcium release by Ins(1,3,4,5)P₄

Research on the role of Ins(1,3,4,5)P₄ as a second messenger has over the years produced conflicting results. Recent evidence (Smith et al., 2000; Hermosura et al., 2000), however, supports the role of InsP₄ in facilitation of InsP₃-mediated calcium release. Whereas Smith et al. (2000) postulate that the function of InsP₄ is attributed to its possible control over ER membrane integrity, Hermosura et al. (2000) show that InsP₄ facilitation arises due to its metabolic effects. The latter view has also been supported in experiments on the Xenopus system (Sims and Allbritton, 1998). Both Ins(1,4,5)P₃ and Ins(1,3,4,5)P₄ share the same dephosphorylating enzyme: InsP 5-phosphatase. This enzyme converts Ins(1,4,5)P₃ to Ins(1,4)P₂ and Ins(1,3,4,5)P₄ to Ins(1,3,4)P₃. As InsP₄ competes with InsP₃ for the common enzyme, rapid InsP₃ degradation is inhibited. Thus, presence of InsP₄ results in a net gain in InsP₃ and hence in facilitation of calcium release.

Our biochemical model has the necessary components involved in interactions between InsP₃ metabolism and calcium release from stores. Hence, it was possible to simulate any effects that InsP₄ might have on calcium release. This is also pertinent in context to our simulations as our model is based on parameters from brain tissue studies that report high expression levels of InsP₃ 3-kinase.

To understand the role of InsP₄, we made two modifications to our model that are represented in Fig. 7 A. Case (i) depicts the reaction scheme for the actual metabolism documented in literature. Here, Ins(1,4,5)P₃ is phosphorylated to Ins(1,3,4,5)P₄ by InsP₃ 3-kinase, and both InsP₄ and InsP₃ are dephosphorylated by the same InsP 5-phosphatase1. Case (ii) is a modification of the basic scheme where we uncouple InsP₃ and InsP₄ dephosphorylation. Here too InsP₃ is phosphorylated by InsP₃ 3-kinase, but only InsP₃ is dephosphorylated by InsP₃ 5-phosphatase1. In this scheme a separate enzyme, InsP₄ 5-phosphatase converts InsP₄ to Ins(1,3,4)P₃. The parameter values for these new phosphatases such as initial concentration and enzyme inhibition by inositol high polyphosphates are identical to those for InsP 5-phosphatase1 in the basic scheme. The K_m and V_max values of these phosphatases for their respective substrates match the corresponding enzyme activities that InsP 5-phosphatase1 exhibits toward InsP₃ and InsP₄ in the actual case.

View larger version (22K): [in this window] [in a new window]   FIGURE 7   Ins(1,3,4,5)P4 metabolically protects Ins(1,4,5)P3 from degradation. (A) Schematic representation of modifications made to the model to analyze the metabolic function of Ins(1,3,4,5)P4. (i) summarizes the actual metabolism documented in literature wherein both Ins(1,4,5)P3 and Ins(1,3,4,5)P4 are dephosphorylated by the same enzyme: inositol phosphate 5-phosphatase1. (ii) Represents a modification made to i, wherein Ins(1,4,5)P3 is dephosphorylated by inositol trisphosphate 5-phosphatase1, and Ins(1,3,4,5)P4 is dephosphorylated by inositol tetrakisphosphate 5-phosphatase. (iii) Represents a model wherein all details of InsP3 metabolism have been replaced by a single step InsP3 degradation to inositol. (B) Responses of InsP3 for case A (i), A (ii), and A (iii) within the non-Osc-model. Stimulus used is a 30-s 0.5 µM pulse of glutamate. InsP3 response for case (i) is 1.8-fold greater and rises 25% faster than responses in cases (ii) and (iii). (C) Responses of InsP3 in case A (i), A (ii), and A (iii) within the Osc-model. Stimulus used is a 2-min 0.5 µM pulse of glutamate. InsP3 response for case (i) is threefold greater than responses in cases (ii) and (iii). InsP3 responses to case A (iii) in both the non-Osc-model and the Osc-model are represented as a gray line.

In case (iii) we replace the entire network of InsP₃ metabolism with a single degradation step from InsP₃ to inositol. The degradation rate was set at 1.75/s, which produces the same basal turnover of InsP₃ as that in case (i) and (ii). The case (iii) model was used to characterize how InsP₃ and calcium dynamics change in a model that does not account for detailed InsP₃ metabolism. A comparison between the simulation outputs in case (ii) and (iii) can also suggest the importance of InsP₄ within the metabolic cascade. If model behavior upon uncoupling of InsP₃-InsP₄ dephosphorylation resembles the behavior upon complete deletion of the metabolic network, it would imply that InsP₄ represents the main effector of the entire network that determines the actual InsP₃ response.

Case (i), (ii), and (iii) models were made for both the non-Osc-model and the Osc-model. A 30-s 0.5 µM glutamate pulse was used to stimulate each of the three non-Osc-models (Fig. 7 B), and a 2-min pulse of similar amplitude was used for each of the Osc-models (Fig. 7 C). We find that InsP₄ indeed protects InsP₃ against hydrolysis. For the non-Osc-model and Osc-model, respectively, the peak InsP₃ response in the presence of the entire InsP₃ metabolic cascade was 1.8-fold and 3-fold greater than the response in the absence of either InsP₃-InsP₄ dephosphorylation coupling or detailed metabolism. The high degree of overlap of the InsP₃ responses for case (ii) and (iii) models suggests that in the amplitude domain, the influence of the detailed metabolism on InsP₃ dynamics is primarily mediated by Ins(1,3,4,5)P₄. Further for the non-Osc-model, the rise time for InsP₃ decreases by ~25% as a consequence of the protective role of InsP₄. Thus, our simulations predict that metabolic effects of InsP₄ modulate the temporal as well as the peak characteristics of the InsP₃ response.

Given the facilitation of InsP₃ buildup by InsP₄, we next wanted to analyze whether this translates into a greater mobilization of calcium. Because in our model we have not incorporated any direct influence of InsP₄ on any calcium channels, a facilitation observed in calcium release would imply a purely metabolic effect of InsP₄. Fig. 8 represents the calcium responses in the non-Osc-model and the Osc-model, that correspond to the InsP₃ responses shown in Fig. 7, B and C. Responses within the non-Osc-model (Fig. 8 A) show that InsP₄ also facilitates the calcium release response via its positive effect on InsP₃. The peak calcium release in the presence of the intact metabolic network was approximately threefold the calcium response in the absence of enzyme competition between InsP₄ and InsP₃. Moreover, the shape of the Ca²⁺ response also changed from a biphasic curve to a bell-shape in the modified versions of the reference model. The rise time for calcium in case (i) was ~60% smaller than that in case (ii) and (iii). Thus, both the increase in amplitude and the decrease in rise time for calcium were greater than those observed for the InsP₃ response (Fig. 7 B) in the non-Osc-model. This suggests that the magnitude of InsP₄ facilitation is greater for InsP₃-mediated Ca²⁺ release than for the metabolic response of InsP₃.

View larger version (21K): [in this window] [in a new window]   FIGURE 8   Ins(1,3,4,5)P4 facilitates Ca2+ release via its metabolic effect. (A) Responses of Ca2+ to cases (i), (ii), and (iii) of Fig. 7 A within the non-Osc-model. Stimulus used is identical to that for Fig. 7 B (0.5 µM glutamate for 30-s). Ca2+ response for case (i) is threefold greater and rises 60% faster than response in cases (ii) and (iii). Ca2+ response to case A (iii) is represented as a gray line. (B-D) Responses of Ca2+ within the Osc-model to cases (i), (ii), and (iii) of Fig. 7 A, respectively. Stimulus used in B, C, and D is identical to that for Fig. 7 C, 0.5 µM glutamate for 2-min (solid bars). Frequency of Ca2+ oscillations is modulated by InsP4. An oscillation frequency of 2/min in B is reduced to 1.5/min in C and D.

We also noted that the temporal responses of InsP₃ and Ca²⁺ upon InsP₃-InsP₄ dephosphorylation uncoupling were almost identical to their behavior upon complete deletion of the metabolic network (Fig. 7 B and 8 A). This implies that InsP₄ is the principal effector of the inositol phosphate cascade that modulates InsP₃ and, subsequently, Ca²⁺. InsP₄ kinetics are mainly governed by its synthesis via InsP₃ 3-kinase. The CaM and CaMKII activated forms of this enzyme have reversible kinetics in our model. If these enzyme activities are modeled using the Michaelis-Menten formulation, responses for case (ii) and (iii) model modifications differ markedly (not shown). For the irreversible enzyme scheme, the large influence of the first order network interaction with InsP₄ on InsP₃/Ca²⁺ is reduced, and secondary network interactions become necessary to explain their responses. However, because reversible enzyme kinetics are a more accurate representation of metabolic flow in this case, we suggest that behavior of InsP₃/Ca²⁺ is governed by the metabolic network mainly at the first rather than higher order interaction level.

Recent experiments (Zhu et al., 2000) show that Ins(1,3,4,5)P₄ can function as a frequency regulator of calcium oscillations in cells. Inhibition of InsP₃ 3-kinase can significantly reduce the oscillation frequency or abolish calcium oscillations, depending on the degree of enzyme inhibition. We also analyzed the effects of InsP₄ on calcium oscillations within our Osc-model. Fig. 8, B, C, and D represent the oscillatory calcium responses that correspond to the case (i), (ii), and (iii) time curves of InsP₃ in Fig. 7 C. The calcium curves show no modulation of amplitude but marked temporal differences. The ineffectiveness of InsP₄ in mediating a change in peak calcium is not surprising. Maximal calcium released per spike is tightly regulated by positive and negative feedback of calcium onto the InsP₃ receptor. Only drastic alterations in the cytosolic or ER calcium buffering would be expected to vary the peak height of calcium per oscillation. At the same time, the frequency modulating effect of InsP₄ can also be explained. Fig. 7 C shows that the peak InsP₃ response for case (i) is threefold greater than the response when InsP₃-InsP₄ metabolism is decoupled. A higher InsP₃ buildup corresponds to faster response kinetics of the InsP₃ receptor (Fig. 3 B). Thus, higher InsP₃ levels generate more frequent calcium spikes within the same time than lower InsP₃ levels. Our simulations also display this phenomenon. The intact InsP₃ metabolism model with coupled InsP₃-InsP₄ dephosphorylation shows maximal oscillation frequency of 2/min (Fig. 8 B). On the other hand, the frequency drops to 1.5/min for both case (ii) and (iii) in which InsP₄ does not protect InsP₃ from hydrolysis (Fig. 8, C and D). Our model does not include any direct Ca²⁺ mobilizing effects of InsP₄. Thus, we suggest that the metabolic function of InsP₄ to protect InsP₃ from hydrolysis is sufficient to generate higher frequency oscillations of calcium.

Function of InsP₄ as a "coincidence detector"

Natural stimulus inputs are often composed of repetitive pulse patterns. It adds to the response capabilities of a cellular system if it can detect stimuli spaced in time, apart from stimuli of different intensities. It has been suggested that InsP₄ may function as a "coincidence detector" for the InsP₃ signaling pathway (Irvine, 2001; Parker and Ivorra, 1991). This means that an otherwise subthreshold stimulus can generate a calcium response if it is coincident with presence of some InsP₄ that has remained in the system from a similar previous stimulation. Hermosura et al. (2000) performed a key experiment to demonstrate this phenomenon. They used a paired pulse stimulus protocol wherein the GPCR agonist concentration used was below the threshold for any measurable calcium release. When the second pulse of agonist followed the first pulse within a particular time frame, an intracellular calcium signal was detected. Such facilitation was not seen when the InsP₃ 3-kinase was pharmacologically blocked.

We investigated whether our Osc-model, which has a more accurate representation of InsP₃ receptor kinetics than the non-Osc-model, can produce facilitation of paired subthreshold stimuli. As shown in Fig. 9, our pulse protocol consisted of an initial 20 nM glutamate pulse for 20-s followed by a second identical pulse after 90, 100, 110, 120, or 130 s. A single 20-nM glutamate pulse of 20-s duration, is incapable of generating a calcium spike. If the second 20-nM pulse succeeds the first within 90 s, a calcium response is always generated. This is because the subthreshold levels of InsP₃ generated in the system by the two stimuli sum to become suprathreshold and release calcium. After 90 s of the first stimulus, InsP₃ levels have significantly reduced. At this time however, InsP₄ levels remain in the system due to its slower degradation than InsP₃ (Fig. 2). If the 20 nM glutamate pulse is now given, this residual InsP₄ can perform its metabolic function to protect any new InsP₃ from degradation. The higher levels of InsP₃ generated during the second pulse would then result in calcium release. Fig. 9 A shows such an effect, wherein the four calcium spikes displayed correspond to stimulus interpulse intervals of 90, 100, 110, and 120 s, respectively. The corresponding response of InsP₃ shows that its levels are potentiated during the second glutamate stimulus (Fig. 9 B). No calcium release is seen for the last stimulus protocol with a pulse spacing of 130 s. By this time, residual InsP₄ from the first stimulus has degraded below the threshold level at which it can protect new InsP₃ from fast hydrolysis. We further confirmed that the paired pulse facilitation was due to the metabolic function of residual InsP₄ during the second pulse, and not simply due to summing of InsP₃ responses across temporally close stimuli. For this, we subjected our version of the Osc-model with decoupled InsP₃-InsP₄ metabolism (Fig. 7 A, case (ii)) to the same stimulus protocol as described above. This model did not show any calcium spiking for the paired stimuli (Fig. 9 D). Our simulations thus show that InsP₄ can serve as a short lived "memory" for a previous exposure of a system to an InsP₃ generating stimulus.

View larger version (20K): [in this window] [in a new window]   FIGURE 9   Protective role of InsP4 leads to paired pulse facilitation. All stimulus pulses (solid bars) are of 20 nM glutamate delivered for 20-s. Pulse 1 initiates at 0 time. Pulse 2, shown as solid bars in a ladder arrangement in A, follows pulse 1 after 90, 100, 110, 120, or 130 s. (A-C) Responses of Ca2+, InsP3, and InsP4, respectively, to the paired pulse protocols. The four Ca2+ spikes in A correspond to delivery of pulse 2 after 90, 100, 110, or 120 s, respectively. An interpulse interval of 130 s does not generate any Ca2+ response. Interpulse intervals less than 90 s (not shown here) produce multiple Ca2+ spikes as residual InsP3 from pulse 1 summates with the new InsP3 from pulse 2. Beyond 90 s, InsP3 from pulse 1 has significantly diminished and Ca2+ release effects during pulse 2 are attributed to InsP4. (D) Response of the Osc-model that has decoupled InsP3-InsP4 dephosphorylation to the same paired pulse protocol. The Ca2+, I