Introduction

Inositol 1,4,5-trisphosphate receptor (IP₃R) is a ligand-gated calcium (Ca²⁺) release channel typically expressed on the endoplasmic reticulum (ER) in neurons and many other cell types. It has a major role in intracellular Ca²⁺ dynamics which, in turn, is involved in many cellular processes such as muscle contraction, neurotransmitter release, vesicle secretion, fertilization, gene transcription, immunity, and apoptosis. In neurons, dynamical changes in Ca²⁺ concentration ([Ca²⁺]) are involved, among others, in neuroplasticity and development (see recent reviews [1], [2]), and in neurodegeneration (see [3], [4]). Transient, repetitive changes in cytosolic Ca²⁺ concentration are crucial for synapse modification and plasticity, including long-term potentiation (LTP) and long-term depression (LTD) [5]–[8]. These phenomena constitute the biological basis for learning and memory formation in the brain [8], [9]. Particularly in the cerebellum, IP₃Rs are relatively highly expressed in Purkinje cells [10]. Ca²⁺ release from ER has been shown to be a key mediator of cerebellar LTD [11].

The inositol 1,4,5-trisphosphate receptor is a tetrameric receptor-channel, consisting of four sub-units. In total, three different genes (ITPR1, ITPR2, and ITPR3) encode three different types (1, 2, and 3) of IP₃R and their splice variants from which homo- or heterotetramers can form [12]. IP₃R is activated and opened by both IP₃ and Ca²⁺. Ca²⁺ can also act as the inhibitor of IP₃R in higher concentrations. IP₃ is produced from phosphatidylinositol 4,5-bisphosphate (PIP) by phospholipase C (PLC). After a cell is stimulated (for example by glutamate in neurons) certain G protein- or tyrosine kinase-linked receptors are activated. These, in turn, can activate PLC. ER acts as a Ca²⁺ store, and while open, IP₃R can release Ca²⁺ from ER lumen to the cytosol. Transient rises or oscillations in Ca²⁺ concentration can then activate various enzymes and even induce changes in the transcriptional level. IP₃Rs are known to be responsible for the phenomenon called Ca²⁺-induced Ca²⁺ release (CICR), in addition to ryanodine receptors (RyRs) [13], [14].

In order to develop models for ion channels and receptors detailed data on the structure and function of the modeled entity is required. The function of IP₃R has been studied with electrophysiological techniques. However, since IP₃Rs are prevalently located on the endoplasmic reticulum of a cell, performing the recordings is not straightforward. The first recordings performed on IP₃Rs involved isolated microsomes from smooth muscle cells incorporated into artificial lipid bilayer [15]. Later, the same technique has been used, for example, for IP₃R in canine cerebellum [16]–[20], in mouse cerebellum [21], and in HEK cells [22] (IP₃R recombinantly expressed). IP₃Rs have also been recorded from the plasma membranes of DT40 cells [23] (IP₃R endogenously expressed (native)) and DT40-3KO cells [24], [25] (stably expressed IP₃R construct, native IP₃R ablated). Since the nuclear membrane is a continuation of the ER, IP₃Rs have also been recorded from isolated nuclei of Xenopus oocytes (for example [26] (recombinantly expressed and native IP₃Rs), Purkinje neurons and granule cells [27], [28] (IP₃R endogenously expressed), and DT40 cells [23], [29]. These kind of data are of great value when developing a model for ion channel kinetics. However, the electrophysiological raw data on IP₃R is not available in any of the publicly available databases, but its statistics is described in publications. For example, the dependence of open probability on cytosolic Ca²⁺ or IP₃ concentrations is given ([16], [19], [20], [29], [30]). In some cases, the open and closed time distributions [18], [20], [22] or mean open time [17], [18], [22], [29] are also reported. In an ideal case, the raw data would be publicly available in a database and a modeler could extract all needed statistical measures out of the data or use the raw data for automated estimation of model parameter values.

In addition to electrophysiological measurements, Ca²⁺ imaging and radioactive assays have also been used to study the behavior of IP₃R in vitro. For example, Fujiwara et al. [31] analyzed the kinetics of Ca²⁺ release via IP₃R in controlled cytoplasmic environment in permeabilized cerebellar Purkinje cells. In addition, superfusion and ⁴⁵Ca²⁺ release assay (radioactive assay) have been used for studying the Ca²⁺ release and inhibition of IP₃R by Ca²⁺ in hepatic microsomes [32]–[34]. These kind of studies give more detailed information on the IP₃R regulation by IP₃and Ca²⁺ and their affinities than electrophysiological studies. In some cases, the data obtained from Ca²⁺ imaging studies or from radioactive assays has been used in modeling studies, for example Fujiwara et al. [31] by Doi et al. [32] and Dufour et al. [35] by Sneyd et al. [36].

In order to reach a better understanding of the dynamical behavior of IP₃R, as well as its involvement in various cellular processes, it is of interest to build models of IP₃R. Computational models are important for understanding the time evolution, dynamics, and regulation of ion channels and intracellular proteins and enzymes [37], [38]. Several models have previously been proposed to describe the behavior of IP₃R (for a comprehensive review, see, for example [39]). There are models presented for different types of IP₃R (type 1, 2, and 3) [12] in different animals, tissues and cells (for example Xenopus oocyte [40], cerebellar cells [41], pancreatic acinar cells [42], and hepatic cells [32]). The first and most well-known model is the one by De Young and Keizer [41]. Some models for IP₃R have been compared either analytically or by means of simulation [36], [43]–[45], and later reviewed [39], [46].

The majority of the existing models is deterministic. Deterministic approaches, however, do not give biologically valid results and are not always capable of modeling the random behavior observed with small numbers of molecules [47]–[50]. Stochastic modeling is therefore more and more used for describing the dynamics of a biochemical system. The stochastic approach is always valid whenever the deterministic approach is valid, but when the deterministic is not, the stochastic might sometimes be valid [51]. Most commonly, deterministic methods and, in some cases, analytical methods are used to investigate the properties of IP₃R models (see, for example [43] or [52]). More rarely, stochastic methods are applied [53], [54], even though it is known that the behavior of ion channels is stochastic.

Despite the wealth of IP₃R models the selection of a specific model for describing IP₃R related calcium dynamics or signaling is not straightforward. The models are seldom generic in nature and capable of describing all possible data obtained for a specific IP₃R or cell type. The reason for this is that the models are developed for some specific purpose, describe the behavior only in certain experimental conditions, or the dynamics are not fully analyzed to validate the model. This can be due to the limited access to experimental data. We therefore wanted to study the dynamics of existing models in detail and to specifically address their suitability in the context of complex neuronal models. In this work, the interest is set on the type 1 IP₃R because it is most commonly expressed in neurons [10]. After a preliminary study, we chose four models [35], [55]–[57] for a more detailed analysis and comparison. Other models did not meet our criteria. The chosen models were originally developed by using data either from IP₃R in canine cerebellum or type 1 IP₃R. As the selected models are biophysically realistic and based on the law of mass action, they can be implemented to the stochastic simulation tool STEPS [58], [59] used in this study. Additionally, we decided to concentrate on computationally inexpensive IP₃R models so that it would be possible to integrate them as part of larger model for calcium dynamics or synaptic plasticity. We validated the functionality of the models by comparing the statistical behavior of IP₃R channel kinetics (open probability curves, mean open times, and open and closed time distributions) to the equivalent obtained by electrophysiological recordings from IP₃Rs expressed in neurons.

Our results show firstly, that the behavior of the studied models varies in similar simulation conditions and, secondly, some models show quite unrealistic kinetic behavior. We therefore conclude that the kinetics of IP₃R (open and closed times and the open probability) with different concentrations of both Ca²⁺ and IP₃ should be more carefully addressed when new models for IP₃R are developed.

Materials and Methods

In our present work, after a preliminary review on existing IP₃R models, we selected four models [35], [55]–[57] for comparison. The selection was based on the following criteria: (1) relative simpicity (i.e. the model should have less than 20 states), (2) development based on data obtained from neuronal or type 1 IP₃R, and (3) basis in the law of mass action (the reactions include binding and unbinding reaction and state transitions). As our ultimate goal is to find a model that can be an integral part of a larger model for Ca²⁺ dynamics or synaptic plasticity in neurons, it is an advantage to have a structurally simple model. The selected models are based on the law of mass action and can thus be implemented into the stochastic simulators such as STEPS [58].

Models

The model of Othmer and Tang.

The model of Othmer and Tang [55] is one of the earliest and small- scaled models regarding the number of states. There is only four states, since the binding order of Ca²⁺ or IP₃ is not free, but sequential, opposite to the models of De Young and Keizer [41] or Bezprozvanny and Ehrlich [18]. Othmer and Tang [55] assume that IP₃ has to bind to its binding site before Ca²⁺ can bind and the channel can open, as well as the activating Ca²⁺ has to bind to its site before the inhibition by Ca²⁺ can occur. The schematic representation of the model of Othmer and Tang [55] in Figure 1A and the parameter values in Table 1 were used in this study. The model of Othmer and Tang [55] has been used before as a part of a larger model for calcium dynamics, for example, by Mishra and Bhalla [60].

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Figure 1. Schematic representation of the states and transitions of the IP₃R models.

(A) Othmer and Tang [55] (forward direction of a reaction is to the right) (B) Doi et al. [35] (forward direction of a reaction is to the right or up), (C) Fraiman and Dawson [57] (forward direction of a reaction is to the right or down) (D) Dawson et al. [56] (forward direction of a reaction is the to the direction of binding a ligand or in the plain state transitions from left to the right).

https://doi.org/10.1371/journal.pone.0059618.g001

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Table 1. Rate constants for IP₃R model of Othmer and Tang [55].

https://doi.org/10.1371/journal.pone.0059618.t001

The model of Dawson et al.

Dawson et al. [56] built a model for IP₃R, using a RyR model by Sachs et al. [61] as their starting point, to understand the adaptive and incremental behavior of IP₃R. The model of Dawson et al. [56] is applicable to type 1 and 2 IP₃Rs and with some modification to type 3. Dawson et al. [56] assume that IP₃R has two conformations, R and P. The conformation R can bind four IP₃molecules rapidly, but with low affinity, to reach an open state. The conformation P, on the other hand, slowly binds four IP₃ molecules, but with high affinity, to reach a closed state where it is thereafter possible to reach the open state. In this work, Scheme 2 from the original paper was used with two exceptions: the flux through an open channel (reactions 14 and 16 in the original paper) and the diffusion of released Ca²⁺ (reaction 17) were not taken into account in order to make the model comparable with other models. This does not have an effect on the actual channel kinetics of the receptor as the removed reactions deal with Ca²⁺ flux and diffusion. Moreover, we used constant Ca²⁺ concentration and the simulated reactions happened in well-mixed system and in the present work only the kinetics of the IP₃R, not Ca²⁺ dynamics was studied. We used the the model presented in Figure 1D and the parameter values given in Table 2.

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Table 2. Rate constants for IP₃R model of Dawson et al. [56].

https://doi.org/10.1371/journal.pone.0059618.t002

The model of Fraiman and Dawson.

The IP₃R model of Fraiman and Dawson [57] was originally built to study the effects of different Ca²⁺ concentrations inside the ER to the kinetics of IP₃R. It is the only model included in the present study that has a Ca²⁺ binding site inside the ER in addition to the cytosolic binding sites. The state scheme of the model of Fraiman and Dawson [57] is presented in Figure 1C and the parameter values used in this work are in Table 3.

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Table 3. Rate constants for IP₃R model of Fraiman and Dawson [57], taken from [67].

https://doi.org/10.1371/journal.pone.0059618.t003

Originally, six states, O_a, O_b, O_c, P_a, P_b, and P_c, were considered open. However, it has been experimentally shown that IP₃R needs IP₃ to reach a stable open conformation [33], [62]. For this reason, we neglected three of the original open states (i.e., they were considered closed) in the present work and only states O_a, O_b, and O_c were considered open. In addition, in the original publication [57], the rate constant of the transition from A₁₀ to A₀₀ is defined as ‘detailed balance’, with no given numerical value. In our study, it was mandatory to have a numerical value for the parameter and thus we fixed the parameter by testing three values with open probability simulations (data not shown). The parameter values of 0 s⁻¹ and 200 s⁻¹ produced identical results which were in accordance with the results in the original publication [57], while the value of 2000 s⁻¹ slightly upraised the left side of the open probability curve. Based on these simulations we chose the value of 200 s⁻¹ for the transition from A₁₀ to A₀₀ (reaction 7, k_b) and concluded that it was in the range of what was originally used.

The model of Doi et al.

The IP₃R model of Doi et al. [35] was originally published as part of a larger model for Ca²⁺ dynamics in the cerebellar Purkinje cell spine to investigate the role of IP₃Rs as a coincidence detector of two input signals. Doi et al. [35] constructed their model based on a conceptual model of Adkins and Taylor [34]. Doi et al. [35] used experimental data by Khodakhah and Ogden [63], Marchant and Taylor [33], and Fujiwara et al. [31] to define the structure and kinetics of the model and experimental data by Bezprozvanny et al. [16] to test how well the model can reproduce the bell-shaped curve. A schematic representation of the model is presented in Figure 1B and the rate constants for each reaction in Table 4. In the model of Doi et al. [35], IP₃R has seven states and the receptor needs to bind both IP₃ and Ca²⁺ to open and thus provide Ca²⁺ flux from ER lumen to cytosol. In this model, IP₃R has one open state, RIC.

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Table 4. Rate constants for IP₃R model of Doi et al. [35].

https://doi.org/10.1371/journal.pone.0059618.t004

Results

We compared four kinetic models previously developed for IP₃ receptor function by simulating them with the Gillespie stochastic simulation algorithm of STEPS simulator. The comparison was done by analyzing the steady-state behavior, such as the open probability, open and closed time distributions, and the mean open and closed time. Here we show that the behavior of the models varies and some models behave somewhat unrealistically.

Open probability

It has been experimentally shown that the open probability () of IP₃R is dependent on the cytosolic Ca²⁺ concentration and that the dependence is bell-shaped [16]. We repeated similar experiments by computational means and tested whether the selected four models are capable of expressing the bell-shaped curve. All the models except the model of Dawson et al. [56] produced the bell-shaped curve (see Figure 2A). Instead, the model of Dawson et al. [56] (blue in Figure 2A) produced an s-shaped curve similarly as in a previous comparison study by Sneyd et al. [36]. The model of Othmer and Tang [55] (green in Figure 2A) reaches the highest ( = 0.33) at cytosolic Ca²⁺ concentration around 80 nM. The model of Doi et al. [35] (magenta in Figure 2A) and the model of Fraiman and Dawson[57] (red in Figure 2A) reach the highest (  = 0.15 and  = 0.38, respectively) around [Ca²⁺]  =  300 nM, which is closest to the experimentally obtained values ([Ca²⁺]  = 250 nM by Bezprozvanny et al. [16] and [Ca²⁺]  =  200 nM by Kaznacheyeva et al. [22]). The absolute value of obtained in simulations cannot be directly compared to the experimental data, because Bezprozvanny et al. [16] and Kaznacheyeva et al. [22] report only normalized values, not absolute values, for .

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Figure 2. Open probability of IP₃R as a function of (A) cytosolic Ca²⁺ concentration (IP₃  = 10 M) and (B) cytosolic IP₃ concentration (Ca²⁺  = 0.25 M).

Green: Othmer and Tang [55], Blue: Dawson et al. [56], Red: Fraiman and Dawson [57], Magenta: Doi et al. [35].

https://doi.org/10.1371/journal.pone.0059618.g002

The open probability of IP₃R is also dependent on cytosolic IP₃ concentration (see for example [17], [27], [29]). The open probability curves of the models obtained in simulations are shown in Figure 2B. All the models except the model of Dawson et al. [56] (blue in Figure 2B) follow the s-shape that is reported in experimental studies [17], [27], [29]. In their study on IP₃Rs on Purkinje cell nuclear membrane, Marchenko et al. [27] have shown that the stays close to 0 until IP₃ concentration reaches 0.3 M and keeps rising until IP₃ concentration is 3 M ([Ca²⁺]  =  .25 M). Watras et al. [17] have shown that the rise starts when IP₃ concentration is 0.03 M and settles after 1 M. The in models of Dawson et al. [56] (blue in Figure 2B) and Doi et al. [35] (magenta in Figure 2B) starts rising approximately at the same IP₃ concentration as in [27], but the elevation does not stop at the right concentrations. In the models of Othmer and Tang [55] (green in Figure 2B) and Fraiman and Dawson [57] (red in Figure 2B), starts rising one or two orders of magnitude too low when compared to the experimental results.

Kaftan et al. [19] have shown in their experiments on cerebellar IP₃R that the bell-shaped Ca²⁺-dependence curve moves upward and to the right when IP₃ concentration is increased. They used IP₃ concentration values of 0.02, 0.2, 2, and 180 M. We used the same concentrations, in addition to their fivefold values, except 180 M in our simulation for all the models (results in Figure 3). The model of Othmer and Tang [55] (Figure 3A) shows a shift upward and to the left, the model of Dawson et al. [56] (Figure 3B) upward, and the models of Fraiman and Dawson [57] (Figure 3C) and Doi et al. [35] (Figure 3D) upward and slightly to the left when IP₃ concentration increases. Similar trend has also been shown for the model of Othmer and Tang [55] by Diambra and Guisoni [66] and Tang et al. [43]. None of the models reproduced the results presented by Kaftan et al. [19].

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Figure 3. Open probability of IP₃R as a function of cytosolic Ca²⁺ concentration in different IP₃ concentrations.

(A) Othmer and Tang [55] (B) Dawson et al. [56] (C) Fraiman and Dawson [57] (D) Doi et al. [35].

https://doi.org/10.1371/journal.pone.0059618.g003

Mean open and closed times and distributions of open and closed times

Bezprozvanny and Ehrlich [18] reported that the mean open time of canine cerebellar IP₃R is 2.9 0.2 ms and Kaznacheyeva et al. [22] that the mean open time of wild-type rat cerebellar IP₃R is 4.20.5 ms and that the open and closed times have exponential distributions (black dashed line in Figure 4E and 4K) in certain experimental conditions (lipid bilayer experiments, [IP₃]  =  2 M, [Ca²⁺]  =  0.2 M). We simulated the selected models in these same conditions (Sim 1, results in Table 5 and Figure 4A–F) and, in order to take into account the affinity difference [31], with five times greater IP₃ concentration (Sim 2, results in Table 5 and Figure 4G–L). The mean open times of the model of Fraiman and Dawson [57] are 2.5 ms (Sim 1) and 2.6 ms (Sim 2). These values are close to the experimentally obtained values. The mean open times obtained with the other models are an order of magnitude smaller (0.5 ms for Dawson et al. [56] and Doi et al. [35]) or significantly greater (460 ms, Othmer and Tang [55]). None of open time distributions of the selected models (Figures 4A-C and 4G-I) follow the experimental distribution by Kaznacheyeva et al. [22] fully, but all give, however, the exponential shape (see Figures 4B and 4K). The open time distribution of the model of Fraiman and Dawson [57] is the closest to experimentally [22] obtained distribution (see Figures 4B, 4H). The same applies also to the closed time distributions (see Figures 4E, 4K).

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Figure 4. Distribution of IP₃R open and closed times for all the selected models obtained in simulation conditions Sim 1 (A–F) and Sim 2 (G–L).

(A) Open time distributions of all the models in conditions Sim 1, (B) Enlarged from A, (C) Enlarged from B, (D) Closed time distributions of all the models conditions Sim 1, (E) Enlarged from D, (F) Enlarged from E, (G) Open time distributions of all the models conditions Sim 2, (H) Enlarged from G, (I) Enlarged from H, (J) Closed time distributions of all the models conditions Sim 2, (K) Enlarged from J, (L) Enlarged from K. Experimental data is from [22]. In simulation conditions Sim 1 [Ca²⁺]  = 0.2 M, [IP₃]  =  2 M and Sim 2 [Ca²⁺]  = 0.2 M, [IP₃]  = 10 M (as shown in Table 6).

https://doi.org/10.1371/journal.pone.0059618.g004

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Table 5. Mean open and closed times of IP₃R of the selected models.

https://doi.org/10.1371/journal.pone.0059618.t005

Moraru et al. [20] have presented open time distributions for canine cerebellar IP₃R in two different conditions (lipid bilayer experiments, [Ca²⁺]  = 0.1 and 0.01 M, and [IP₃]  = 2 M) (black dashed line in Figures 5 and 6). We simulated the behavior of the selected models in these same experimental conditions (Sim 3 and Sim 4, results in Table 5 and Figure 5) and also with fivefold IP₃ concentration (Sim 5 and Sim 6, results in Table 5 and Figure 6). The distributions in the wet-lab experiments are of exponential shape [18]–[20], [22] and simulation results also show exponential shape for all the models. The only distributions that are also otherwise similar to the ones obtained in wet-lab experiments by Moraru et al. [20] are the distributions of the model of Fraiman and Dawson [57] (Figures 5B, 5H, 6B, and 6H). All the simulation conditions used are summarized in Table 6.

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Figure 5. Distributions of IP₃R open and closed times for all the selected models obtained in simulation conditions Sim 3 (A–F) and Sim 4 (G–L).

(A) Open time distributions of all the models in conditions Sim 3, (B) Enlarged from A, (C) Enlarged from B, (D) Closed time distributions of all the models in conditions Sim 3, (E) Enlarged from D, (F) Enlarged from E, (G) Open time distributions of all the models conditions Sim 4, (H) Enlarged from G, (I) Enlarged from H, (J) Closed time distributions of all the models conditions Sim 4, (K) Enlarged from J, (L) Enlarged from K. Experimental data is from [20]. In simulation conditions Sim 3 [Ca²⁺] = 0.1 M, [IP₃] = 2 M and Sim 4 [Ca²⁺]  =  0.1 M, [IP₃]  = 10 M (as shown in Table 6).

https://doi.org/10.1371/journal.pone.0059618.g005

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Figure 6. Distribution of IP₃R open and closed times for all the selected models obtained in simulation conditions Sim 5 (A–F) and Sim 6 (G–L).

(A) Open time distributions of all the models conditions Sim 5, (B) Enlarged from A, (C) Enlarged from B, (D) Closed time distributions of all the models conditions Sim 5, (E) Enlarged from D, (F) Enlarged from E, (G) Open time distributions of all the models conditions Sim 6, (H) Enlarged from G, (I) Enlarged from H, (J) Closed time distributions of all the models conditions Sim 6, (K) Enlarged from J, (L) Enlarged from K. Experimental data is from [20]. In simulation conditions Sim 5 [Ca²⁺]  = 0.01 M, [IP₃]  = 2 M and Sim 6 [Ca²⁺]  = 0.01 M, [IP₃]  = 10 M (as shown in Table 6).

https://doi.org/10.1371/journal.pone.0059618.g006

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Table 6. Ca²⁺ and IP₃ concentration used in different simulations for open and closed time distributions.

https://doi.org/10.1371/journal.pone.0059618.t006

The Ca²⁺ concentrations used in the experiments by Moraru et al. [20] are unfortunately at the border or smaller than those observed in a neuron at resting level (i.e., Ca²⁺ used is 0.1 M or less). As IP₃R is, however, known to have functional significance only above the resting level concentrations, more emphasis should be put on physiological conditions in experimental work in the future. In other words, experimental work should additionally be performed with Ca²⁺ concentrations above the known resting level.

Discussion

In this work, four models of IP₃R [35], [55]–[57] were selected among many to examine their steady-state and time series behavior and compare them with experimental data available in literature. We implemented and simulated the selected models using stochastic simulation software STEPS in order to obtain similar data as in single-channel patch-clamp recordings. The open probability curves and statistics, such as the mean open time and open and closed time distributions, were compared to experimental ones obtained in the same conditions. To our knowledge, this is the first detailed evaluation of IP₃R model kinetics with stochastic methods. Our comparative study shows significant differences in the behavior and kinetics of the studied models.

Based on our results, the statistical properties of the model of Fraiman and Dawson [57] seem to be the most similar to the ones obtained in wet-lab experiments. The properties of the model of Othmer and Tang [55] are very different when compared to the experimental data. All the models except the model of Dawson et al. [56] produce the bell-shaped open probability curve for Ca²⁺-dependence and the s-shaped open probability curve for IP₃-dependence as seen in the electrophysiological experiments (for example [16], [17], [27]). However, none of the models reproduce the experimental finding presented by Kaftan et al. [19], which shows that Ca²⁺-dependent open probability curve moves to the right and upward when IP₃ concentration increases. This kind of behavior is shown in the original article by Fraiman and Dawson [57]. The reason why the simulation of the same model in this study did not produce similar behavior might be the slight modification that we were forced to make to the model (defining a numerical value for the one parameter that was originally defined as ‘detailed balance’ and neglecting three of the six open states). It is also notable that there is an Errata [67] published for the original article [57] and that we used the parameter set in the Errata [67].

The simulated open and closed time distributions of all the models follow the exponential distribution as does the data from experiments [18]–[20], [22]. However, the distributions are not similar apart from the distribution of Fraiman and Dawson [57]. The reason for this may be the relatively simple structure of the models, insufficiency of modeled states to reproduce the kinetics, and parameter values that do not fit the data.

According to our results, the mean open time of model of Doi et al. [35] is not congruent with the experimental findings. However, the shape and peak value of the open probability curve are in accordance with experimental data. As the model of Doi et al. [35] has originally been published as part of a larger signal transduction model for LTD induction, some inaccuracy in the behavior of the model could have been corrected by other parameters, such as the Ca²⁺ flux rate and thus the small mean open time does not invalidate the results in the original publication.

As our comparative study points out significant differences in the behavior and kinetics of the studied models, it is of interest to consider reasons for it. We identify four major reasons why the selected models behave differently to each other: 1) the structure (i.e. the equations) and parameter values differ between the models, 2) experimental data that was used in the model development vary, 3) different data handling procedures have been used when developing the models, and 4) model developers did not use automated parameter estimation methods. Next, we will discuss each issue in detail.

Firstly, the most obvious reason for differences in the behavior of models is the structure and parameter values of the models. All the models studied here have different number of states, but this does not cause the differences as such. More importantly, different parameter values and thus the affinities of IP₃, as well as activating and inactivating Ca²⁺, vary between the models. Since the models of Othmer and Tang [55] and Doi et al. [35] reproduce the correct shapes for the open probability curves, re-estimation of their parameters might improve the fitting of models to experimental data. As a general conclusion, all studies neither report the values of all parameters used in simulations nor make it evident which parameter set is used to produce specific results. This makes it difficult to reproduce results (see also discussion in [68]).

Secondly, another reason for the differences in the behavior of the models could be related to the variability in the use of experimental data when constructing the original model. Although the statistical properties of channel kinetics, such as the mean open time and the distributions of open times, are known to be important in properly reconstructing receptor-ion channel kinetics, they are relatively rarely used in developing or evaluating models for IP₃R. Furthermore, there exists a clear difference on how experimental data is used to construct (i.e., to define the structure, the number of states, and the number of parameter values in the model) and fine-tune the models (estimation of the unknown parameters). We have noticed that it is not always clear which data is used in modeling and, particularly, how it is used. In general, the models presented for IP₃R are constructed based on only some of the data or knowledge obtained from various animal species and experiments. Furthermore, data on kinetics of IP₃R have been obtained from various sources: native and recombinantly expressed receptors in cell lines and Xenopus oocytes, and from vertebrate cerebellum or hepatocytes.

Doi et al. [35] use the model of Adkins and Taylor [34] as their starting point and construct the model based on data by Marchant and Taylor [33] and use the open probability curve of Bezprozvanny et al. [16] to study the fitness of their model. The model of Othmer and Tang [55] is also shown to fit the data by Bezprozvanny et al. [16] in addition to data by Watras et al. [17] in [43], but this study does not take the difference in IP₃ affinity [31] into account as Doi et al. [35] or study the open or closed time distributions of the model. Fraiman and Dawson [57] and Dawson et al. [56] use several experimental observations when constructing their model, but they do not report using any data for actual fitting of the model parameters. The data that Dawson et al. [56] compare their model to is more dealing with temporal aspect of Ca²⁺ release and accumulation of Ca²⁺ to cytosol than actual channel kinetics.

Thirdly, the differences between the simulated and experimentally observed open time distributions and mean open times might also be due to differences in data handling procedures. Experimentally observed open time distributions can be biased due to the limitations and established practices regarding the temporal resolution in the patch-clamp recordings, while in the simulations in this study all the events are recorded exactly at the time they happen. Usually the time resolution in patch-clamp recordings is around 1 ms and thus any opening shorter than that would stay unnoticed or be merged with other channel openings.

Fourthly, to our knowledge, automated parameter estimation methods have not been used in the development of the four models here compared. Studies on IP₃R models consider, to some extent, the kinetic ion channel data to define the mathematical structure of the models. However, only a few previous studies use automated parameter estimation techniques and statistical data on ion channel kinetics to fine-tune the IP₃R models [36], [69]–[72].

One of the major challenges in modeling the IP₃Rs is the lack of access to original raw data, for example from electrophysiological measurements, that could be used in quantitative modeling. This data is not currently available in any public database and as the years pass by it becomes extremely hard to acquire the data from its original sources. This problem is not new or limited just to measurements of ion channels but to all neuroscience data [73], [74]. Some suggestions to improve the situation have been made. For instance, De Schutter [75] suggests that data publishing should be distinguished from paper publishing. Furthermore, Ranjan et al. [76] have established an information management framework for ion channel information, which hopefully will make IP₃R experimental data more accessible in the future.

Despite several shortcomings in the development and presentation of models, previous models on IP₃R, including the present comparative study on four stochastic IP₃R models, will give a good setting for constructing a new, realistic model of IP₃Rs for compartmental modeling of neuronal functions. It will be a challenge to develop computationally inexpensive models that can produce realistic stochastic behavior of an individual ion channel. A wealth of evidence indicates, however, an important role of randomly opening ion channels on the global behavior of cells. For example, in neurons the stochastic openings of single ion channels shape the integration of local signals in dendrites or spines [77], stochastic openings of voltage-gated ion channels have an important role in adjusting the transmembrane voltage dynamics [78]–[80], and the reliability of action potential propagation along thin axons is affected by the stochastic opening of voltage-gated ion channels [81]. Furthermore, molecular noise of single ion channel is shown to be translated into global cellular processes in astrocytes [82].

In summary, the development of new IP₃R models clearly calls for both steady-state and kinetic data. Fitting of the new computational models should be done using automated estimation techniques, possibly using Bayesian approaches [72], [83]–[85]. Data for model construction and fine-tuning would ideally be acquired from the same neuronal type as the model is built for.

Acknowledgments

The authors thank Erik De Schutter and Stefan Wils for discussion and help in the use of STEPS in the early phase of the work.