Data-driven IP{}_{3}R modelling

Data-driven modelling of the
inositol trisphosphate receptor (IPR) and its role in
calcium induced calcium release (CICR)

Ivo Siekmann, Pengxing Cao,
James Sneyd & Edmund J. Crampin
Systems Biology Laboratory, Melbourne School of Engineering, University of Melbourne, Australia Department of Mathematics and Statistics, University of Melbourne, Australia Department of Mathematics, University of Auckland, New Zealand School of Medicine, University of Melbourne, Australia
\enddoc@text

1. Introduction

A number of models have been published that relate different physiological processes involving glial cells to calcium dynamics. De Pittà et al. [22] give an overview of current problems in the modelling of astrocytes. One area of continuing interest is the propagation of signals between astrocytes via intercellular calcium waves. Höfer et al. [42] investigated the spreading of signals between astrocytes via calcium waves based on a model by Sneyd et al. [76]. Bennett et al. [8, 7] developed a more detailed model of calcium waves that combines underlying calcium dynamics with ATP release by purinergic receptors in order to demonstrate that calcium waves depend on ATP release rather than on IP diffusion through gap junctions as in the model by Höfer et al. [42]. Edwards and Gibson [27] later published a model that included both modes of signal propagation and concluded that both were necessary to account for data collected from the retina. Recently, the study of calcium waves has been extended from one- or two-dimensional to three-dimensional spatial domains [45]. Macdonald and Silva [50] model wave propagation on an astrocyte network derived from experimental data. The Bennett et al. model was used for investigating spreading depression, a wave of electrical silence that propagates through the cortex and depolarises neurons and glial cells [9].

A fundamental problem in calcium dynamics in general is the question how multiple signals can be encoded by the dynamics of a single quantity, the concentration of calcium. De Pittà et al. [24, 20, 23] investigated how a stimulus could be encoded via the frequency or the amplitude or both frequency and amplitude which demonstrates that two different signals can be represented independently in an individual calcium signal. Dupont et al. [26] showed in a detailed model how the signal received by a particular glutamate receptor is encoded via calcium oscillations.

Lavrentovich and Hemkin [46], Zeng et al. [87], Riera et al. [62], Riera et al. [63] investigated spontaneous calcium oscillations in astrocytes and Li et al. [47] explored their role in spreading depression.

Also the coupling of astrocyte network with the neural network has been investigated. At the single-cell level, De Pittà et al. [21] modelled the interaction of an astrocyte with a synapse. Allegrini et al. [1], Postnov et al. [59] study the influence of a network of astrocytes on a neural network.

Most recently, Barrack et al. [5, 6] explored the role of calcium signalling in neural development. By coupling calcium dynamics with a model of the cell cycle they examine how glial progenitors differentiate to neurons triggered by a calcium signal.

This review of the modelling literature on glial cells clearly demonstrates that the importance of calcium dynamics is well recognised—the majority of studies in the literature accounts for calcium signalling and often models are used to find a link of physiological processes with calcium signalling. In many cell types including glial cells the inositol trisphosphate receptor (IPR) plays a crucial role in inducing oscillatory Ca signals. In the presence of IP, opening of IPR channels leads to Ca release from the endoplasmic reticulum (ER), an intracellular compartment with a very high Ca concentration a few orders of magnitude higher than that of the cytoplasm. The IPR is activated by Ca so that such a release event dramatically increases the open probability of the IPR which induces further release of Ca(henceforth called calcium-induced-calcium-release, or CICR) until a high Ca concentration in the channel environment eventually inhibits the IPR.

The Li-Rinzel model [48], an approximation of the classical De Young-Keizer model [25], is by far the most commonly used representation of the IPR in models of glial cells. Only Allegrini et al. [1] and Lavrentovich and Hemkin [46] chose different models based on Atri et al. [3] or Tu et al. [82], respectively. Dupont et al. [26] use the model by Swillens et al. [79] that explicitly accounts for the effect of interactions in a cluster of IPR channels. Early models of the IPR were designed to account for the bell-shaped Ca dependency of the open probability  of the channel described by Bezprozvanny et al. [10]. Since then the dynamics of IPR in response to varying concentrations of IP, Ca and ATP has been characterised much more comprehensively as well as the differences between the different isoforms of the IPR (among the models mentioned above, in fact, only Tu et al. [82] accounts for the fact that astrocytes predominantly express type II IPR).

The scope of current data-driven models of ion channels has advanced beyond representing the average open probability . Recent models capture the stochastic opening or closing of single IPRs in aggregated Markov models i.e. instead of only modelling the stationary behaviour of the channel they represent the dynamics of the IPR (Section 3.4). Accurate representation of IPR dynamics depends on various sources of experimental data (Sections 3.1-3.2) as well as appropriate statistical methods for fitting Markov models to these data (Section 3.5). Statistical methods automate the process of estimating parameters for a given Markov model. Thus, the main challenge of data-driven ion channel modelling is to define the structure of a Markov model which allows the integration of various sources of experimental data. We illustrate this process with two recent examples of models for the IPR (Sections 3.6 and 3.7).

Once a model for a single channel has been developed, data from small clusters of channels can be used to determine how well the behaviour of a cluster is represented by an ensemble of single-channel models (Section 4.1). Studying the influence of an IPR model on calcium dynamics allows us to evaluate the relative importance of different aspects of single-channel dynamics. Cao et al. [14] showed that the essential features of calcium dynamics in airway smooth muscle could be preserved after iteratively simplifying the IPR model by Siekmann et al. [73] to a two-state model that only accounted for the switching between the inactive “park” and the active “drive” mode. In Section 4.2 it is shown that this also applies to the puff distribution. This demonstrates that modal gating is the most important regulatory mechanism of the IPR. It also emphasises that data-driven modelling of ion channels does not necessarily have to lead to detailed models based on complicated model structures but rather can be used so that relevant data is selected to represent ion channels at the appropriate level of complexity for a given application.

2. Mathematical models of calcium dynamics/CICR

The purpose of a mathematical model of CICR is to explain the emergence of complex intracellular calcium dynamics such as oscillations as the result of interdependent calcium fluxes. This comprises both fluxes into and out of the cell as well as the exchange between the cytosol and intracellular stores (Figure 1).

Figure 1. General structure of calcium fluxes in glial (and other non-excitable) cells. The central component is the flux  through the inositol trisphosphate receptor (IPR). The IPR is activated by binding of IP which is generated upon stimulation of the cell by an agonist. This causes the release of Ca from the endoplasmic reticulum (ER) to the cytoplasm. The resulting elevated Ca concentration increases the open probability of the IPR and the ryanodine receptor (RyR) which stimulates further Ca release. This mechanism is known as calcium induced calcium release (CICR). At high concentrations, Ca inhibits the IPR, i.e. the open probability of the IPR decreases. In consequence, influx into the ER through the SERCA pump dominates the efflux through IPR and RyR so that Ca is reabsorbed by the ER. Ca exchange with the extracellular space is controlled by uptake through various channels () and by extrusion via pumps ().

The dynamics of cytosolic () and stored calcium () resulting from these fluxes can be represented by a system of differential equations:

(1)
(2)

Here, is calcium influx from the extracellular space via calcium channels located in the cell membrane, accounts for calcium removed from the cell by the plasma membrane pump. and  represent calcium release from the endoplasmic reticulum (ER) through the IPR and the RyR, respectively, and  stands for reuptake of calcium into the ER by the SERCA pump. The conversion factor , the ratio of the cytoplasmic volume to the ER volume, is necessary because calcium concentrations are calculated with respect to the different volumes of these two compartments. The model (1), (2) provides a description of Ca concentrations across the whole cell. This means that we cannot account for spatial effects due to heterogeneities of the spatial distribution of IPR, SERCA and other relevant components of the system. By using a deterministic model we further assume that the various Ca fluxes can be described as deterministic after averaging over a large number of channels and transporters. In Section 4 we will consider a stochastic model over a small spatial domain for a cluster of interacting IPRs.

In a whole-cell model of calcium dynamics such as (1), (2), a representation of the IPR must, in principle, just provide a functional expression for

(3)

the ligand-dependent flux through IPR channels present in a cell. Because the calcium concentration  is time-dependent, varies over time. In the early days of modelling of the IPR, phenomenological models were used for representing the IPR flux. A good example is the model by Atri et al. [3]:

(4)

where , and  is the number of open channels. The model by De Young and Keizer [25] is derived from more detailed assumptions on chemical interactions of the channel with its ligands. In Section 3.6 we present a more recent model [84] that is representative for this approach. The Hill function-type terms in (4) enabled Atri et al. to interpret their model in terms of a physical process but the main motivation of the model was to obtain a fit of the calcium-dependent whole-cell flux  to data collected by Parys et al. [58]. From a purely mathematical point of view, phenomenological models seem to be the ideal approach for investigating the role of IPR in calcium dynamics—restriction to minimal models that generate the desired behaviour ensures that model behaviour can be analysed to a great extent. This allows us to test hypotheses on IPR regulation in an elegant way.

But the capability of simple mathematical expressions for the macroscopic flux to perform the appropriate functional role in calcium dynamics is only a relatively indirect test for IPR models. By following a phenomenological approach we mostly ignore data that gives more direct information on the IPR, such as, for example, the molecular structure of the channel protein which can be obtained from crystallography and time series of opening and closing of a single channel from patch-clamp recordings. Taking into account these data may allow us to restrict the set of theoretically possible mathematical expressions and, in this way, also the set of possible mechanism.

3. Data-driven modelling of single IPRs

Because most biophysical data relate to single channels, data-driven modelling involves an important conceptual step—instead of directly specifying the whole-cell flux , we first construct a model for the flux through a single channel. Whereas for the macroscopic flux  which is averaged spatially over many channels distributed across the whole cell the deterministic model (3) is appropriate, representing the flux through a single channel requires a stochastic model. In a second step, is then derived by appropriately averaging over the stochastic fluxes through individual channels.

In Sections 3.1 and 3.2 we describe two sources of data that are commonly used for the construction of ion channel models. Ca release data from small clusters of IPR, so-called calcium puffs (Section 3.3), can be used for validating models of single channels. In Section 3.4 aggregated continuous-time Markov models, the mathematical framework common to all models based on single-channel data, is introduced. A short review of statistical approaches for fitting Markov models to single-channel data is given in Section 3.5. In Sections 3.6 and 3.7 examples of two recent models of the IPR are given in order to illustrate different modelling approaches. Earlier models have been reviewed by [38] and Sneyd and Falcke [77]. Model comparisons [78, 40] generally show that models not parameterised by fitting to experimental data may not do a very good job at reproducing the statistical properties of ion channel kinetics.

3.1. Molecular structure

The mathematical structure of many ion channel models is designed to mimic the chemical structure of the channel protein. The motivation for this approach is to link molecular structure of the ion channel to its function.

In vertebrates there exist three different genes encoding three different types of the IPR. In mammals, type I IPR is ubiquitously expressed but most cells express more than one isoform. The predominant isoform in astrocytes is type II IP[69, 43]. For each isoform there are several splice variants.

Imaging the three-dimensional structure of the complete IPR and RyR channel proteins is challenging and only recently have accurate 3D visualisations of complete IPRs using electron cryomicroscopy (cryo-EM) become available [49]. Parts of the channel can be imaged at higher resolution by crystallography and be superimposed on cryo-EM images [29]. These studies have revealed that IPR channels are tetramers i.e. formed by binding of four IPR proteins. These tetramers may consist of different IPR subtypes but experimental studies have so-far concentrated on investigating homotetramers formed by four copies of the same subtype (but see Alzayady et al. [2]). The classical model by De Young and Keizer [25] took into account this information by building a model from identical subunits that all had to be in an open state for the channel to open (although the de Young-Keizer model assumed three instead of four subunits).

Analysis of the amino acid sequence by mutation experiments have assigned functional roles to various segments, for example, the IP binding core (IBC) which contains an IP binding site has been identified. There is less information on the number and localisation of Ca binding sites. Because localisation of Ca binding sites by mutation studies has been difficult, Foskett et al. [31] infer various Ca binding sensors from the observed co-regulation by IP and Ca, see Foskett and Mak [30] for a summary. Often models assume a certain number of IP and Ca binding sites and represent binding and unbinding of these ligands as transitions between states regulated by mass action kinetics. This modelling approach will be described in more detail in Section 3.6.

3.2. Patch-clamp recordings

Detailed studies of individual ion channels became possible due to the development of the patch-clamp technique. Neher and Sakmann [56] were the first to detect the flow of ions through a single ion channel by measuring the resulting current at constant voltage. The time-course of opening and closing can be inferred from the detected current which stochastically jumps between zero (closed) and one or more small non-zero current levels in the pA range (open) whose sign depends on the valence of the ion and the direction of the current.

Mak and Foskett [53] recently reviewed the single-channel literature of IPR channels. An important experimental development that they highlight relates to the difficulty that IPRs are naturally localised within cells rather than in the cell membrane. Whereas in earlier patch-clamp experiments, IPR channels were studied in artificial lipid bilayers, more recently investigating IPR in isolated nuclei is favoured because it is assumed that nuclei provide an environment similar to the endoplasmic reticulum (ER), the native domain of the IPR.

3.2.1. Stationary data

If ligand concentrations (such as IP, Ca and ATP) are kept constant for the whole duration of the experiment we obtain stationary data. These data allow us to observe the “typical” channel dynamics for a given combination of ligands. The reason that we refer to these data as “stationary” is that we assume that the channel has fully adjusted to the concentration of ligands—the term stationary suggests that the channel has reached its stationary probability distribution, see Section 3.4. Because the stationary solution is only reached asymptotically we can, in theory, never be sure that our ion channel has actually reached equilibrium. Instead we can check if a data set is not stationary by using indicators such as the open probability. If the open probability averaged over a sufficient number of data points spontaneously changes (which indicates the switching of the channel to a different activity level) the channel may exhibit modal gating.

3.2.2. Modal gating

Spontaneous switching between different levels of channel activity at constant ligand concentrations has been observed for a long time. The earliest example is perhaps from a classical study of the large-conductance potassium channel (BK) [52, 51]. In IPR channels modal gating was discovered only relatively recently [44]. The authors found three different modes characterised by high (H), intermediate (I) and low (L) levels of open probabilities, , and . They also realised the importance of modal gating for IPR regulation: they observed that the same three modes seemed to exist for different combinations of ligand concentrations. Because the IPR mostly seemed to adjust the time spent in each of the three modes they proposed that modal gating is the major mechanism of ligand regulation in IPR channels.

One reason that the significance of modal gating has not been appreciated until recently is due to the fact that switching between different modes cannot always be recognised easily without statistical analysis. Recently, Siekmann et al. [72] developed a statistical method which for a given set of single-channel data detects switching between an arbitrary number of modes  characterised by their respective open probabilities . A software implementation which is publicly available under https://github.com/merlinthemagician/icmcstat.git was applied to a large data set from Wagner and Yule [86]. Siekmann et al. [72] found that the same two modes, an inactive “park” () and an active “drive” mode (), were found across all combinations of ligands. There may be various reasons why two modes were observed rather than the three modes found in the earlier study [44], see Siekmann et al. [72] for more details. But more importantly, a detailed study of a bacterial potassium channel (KscA) [17, 16, 15] strongly suggests that the stochastic dynamics characteristic for each mode may be closely related to distinct three-dimensional configurations (conformations) of the channel. Thus, whereas it is often difficult to relate individual open or closed states in ion channel models to distinct conformations of the channel protein, the set of model states that represents a particular mode may, in fact, have a biophysical counterpart [72]. In order to confirm this hypothesis, more studies of modal gating for a variety of channels are needed.

Independent from its biophysical significance, appropriately accounting for modal gating is crucial from a modelling point of view. As we will see in Section 3.4, the phenomenon of modal gating demonstrates that a Markov process must be observed for a sufficiently long time in order to infer the correct stationary distribution, otherwise we observe a “quasi-steady state”. For example, a channel whose kinetics is restricted to an active and an inactive mode can produce intermediate activity only by switching between both modes. Thus, a model that is not capable of switching between different levels of activity is misleading because it produces a constant open probability instead of alternating between highly different open probabilities. In their recent review Mak and Foskett [53] explicitly recognise the importance of modal gating which so far has only been represented in the most recent models [84, 73].

3.2.3. Response to rapid changes of ligand concentrations

Modal gating is an aspect of stationary data collected at constant concentrations of ligands. In contrast, Mak et al. [54] designed an experiment where IP and/or Ca concentrations in the channel environment were rapidly altered in order to simulate an instantaneous change of ligand concentrations. Switching from ligand concentration where the IPR is inactive to conditions where the channel is maximally activated (and vice versa), enabled Mak et al. [54] to investigate the question how fast the IPR responds to varying ligand concentrations. To illustrate the experiment let us consider the change from inhibitory to activating conditions. At an inhibitory condition, the open probability of the channel is very close to zero so that initially the IPR is most likely closed. When changing from an inhibitory to an activating condition the channel will activate but it needs a certain time to respond to the change. In order to measure this latency, Mak et al. [54] recorded the time the channel took from when they altered the ligand concentration until the first opening. For the opposite change from activating to inhibitory conditions they analogously detected the time the channel needed to switch from a high to a low level of activity. This experiment was repeated multiple times for switching between the same conditions which enabled the authors to investigate the latency statistics. It was not only discovered that for some conditions the latencies were surprisingly long but interestingly, they also found that for some conditions the latency distributions were multi-modal which shows that multiple timescales may be observed for the same latency.

Due to the substantial effort required to perform these experiments (which have to be repeated multiple times for each condition where each repeat only gives a single data point rather than a time course) it is unsurprising that these data are very rare. In fact, to date, Mak et al. [54] is the only data set of this kind currently available. Mak and Foskett [53] explain that their data suggests that there may be long refractory periods between release events from the same IPR which makes these results particularly relevant for the modelling of Ca puffs.

3.3. Calcium puffs

So far we have only considered data recorded from single IPRs. In order to understand how the macroscopic flux  arises from the release of many individual channels we have to consider the hierarchical nature of Ca release. As reviewed by Parker et al. [57], Falcke [28], Thurley et al. [80] stochastic opening of a single IPR channel leads to a localised Ca release event (a Ca blip). Such a release further sensitises neighbouring IPR to induce more Ca release through a few tightly clustered IPR by CICR (a Ca puff). Sufficiently many puffs could eventually trigger a global elevation of  that is able to propagate through the entire cell (a Ca wave) [55]. Thus, Ca puffs play a crucial role that: not only are puffs essential for the formation of functional global Ca signals [12] but they also reflect the quantal Ca releases by stochastic openings of IPin vivo [75].

Experimentally, Ca release at a specific spatial position can be initiated by triggering release of caged IP using a laser. A relative measure for the local Ca concentration is obtained by detecting fluorescent dye bound to Ca using a light microscope. For a given point within the cell the resulting time series is characterised by a sequence of stochastic spikes that are highly variable as far as the spike amplitude, the frequency and the time interval between subsequent spikes, the inter-puff interval, is concerned. From a modelling point of view, these data can be used to test wether the single-channel behaviour represented in a model is able to account for the release from a cluster of interacting IPRs. As explained in Section 4.1, Cao et al. [13] found that the original model by Siekmann et al. [73] was incapable of generating the correct stochastic puff distribution as long as the adaptation to different ligand concentrations was assumed to occur instantaneously. After augmenting the model so that it accounted for the latency data by Mak et al. [54] presented in the previous section the puff statistics could be reproduced accurately.

The only other model that accounts for latency data is the model by Ullah et al. [84]. Because the models by Siekmann et al. [73], Cao et al. [13] and by Ullah et al. [84] are the only models that account for all aspects of single-channel data assumed to be necessary for an understanding of the IPR we focus on these two models and the alternative modelling approaches that they represent in Sections 3.6 and 3.7.

3.4. Aggregated continuous-time Markov models

The most natural model for the stochastic process of opening and closing of a single ion channel is the aggregated continuous-time Markov model. A good introduction to the theory reviewed here is the classical paper by Colquhoun and Hawkes [18] which also gives some simple but illustrative examples.

An aggregated continuous-time Markov model is a graph on a set of  closed and open states  (Figure 2).

(a) Ullah et al. [84]
(b) Siekmann et al. [73]
Figure 2. Examples for recent Markov models of the IPR.

Between adjacent states  and  the transition rate (from  to ) is given by  so that the whole model is represented by a matrix with constant coefficients, the infinitesimal generator . The time-dependent probability distribution  over the state set  is the solution of the differential equation

(5)

The stochastic interpretation of (5) is as follows: for a given point in time, one particular state  of the model is “active”. But how long it will take until the current state  is vacated and which state  will be active after a time  cannot be answered with certainty (deterministically) due to the stochastic transitions between states.

For the model defined by (5) the Markov property holds both for the stochastic sequence of active states as well as for the time that it takes until the active state is left.

  1. Which state  will be the next active state only depends on the currently active state , not on previously active states.

  2. The time  it takes until the model exits from the state , also called the sojourn time in , does not depend on the time already spent in .

The second point implies that sojourn times  must be exponentially-distributed because the exponential distribution is the only continuous probability distribution with this property. This explains why multiple open and closed states may be needed for accurately representing the opening and closing of ion channels.

In order to ensure that  is a stochastic vector i.e. for all , the matrix  must be conservative, i.e. for the diagonal elements  we have

(6)

Provided that (6) holds, the solution

(7)

is a stochastic vector for all  if and only if the initial distribution  is a stochastic vector. From (7) the time-dependent open probability of the channel can be calculated by summing over the individual probabilities of all open states.

For large times  the solution  approaches a stochastic vector  which is known as the stationary distribution. This means that provided we wait sufficiently long, the expected frequency of observing a state  approaches a probability . Because  is the solution of a differential equation, is, in fact, a stationary solution of (5) i.e. can be obtained by solving the equation

(8)

This homogeneous linear equation has non-trivial solutions because the matrix  is singular by (6). An argument based on Perron-Frobenius theory for non-negative matrices ensures that  is a unique strictly positive stochastic vector. Moreover, is stable so that for  indeed  approaches  i.e. we have  [68].

3.5. Estimation of Markov models from experimental data

Whereas the mathematical framework of aggregated Markov models was developed a short time after single channel data became available, the statistical estimation of these models is a topic of current research. Most commonly used are approaches based on Bayesian statistics. For a given time series  of open and closed events recorded from an ion channel the conditional probability density , known as the posterior density in the Bayesian framework, is used for determining a suitable Markov model with infinitesimal generator . Note that both  and  are considered as random variables, thus the posterior distribution quantifies how likely a model  is under the condition that data  have been observed. Direct calculation of the posterior  is analytically intractable and computationally prohibitive but efficient approaches for maximum likelihood estimation (MLE) i.e. estimating

(9)

were published in the 1990s [60, 61, 19]. Software implementations of these methods have been made available freely for academic use. Currently, the methods by Qin et al. [60, 61] can be obtained under the name QUB as standalone GUI applications at http://www.qub.buffalo.edu/. DCPROGS based on Colquhoun et al. [19] is still under active development and the source code of the most recent version has been published on github: https://github.com/DCPROGS.

An alternative approach to maximum likelihood estimation has been pursued since the late 1990s. The aim of Markov chain Monte Carlo (MCMC) is to approximate the posterior density  by sampling. MCMC enables us to randomly generate a sequence  of models such that the expected frequency of a model  within this sequence is as large as the density . Thus, by generating a sufficient number of samples, the posterior  is approximated.

The early method by Ball et al. [4] for estimation of a Markov model  depends on a suitable idealisation of discretely sampled measurements to continuous open and closed times. This leads to a difficult statistical problem that has been discussed widely in the ion channel literature as the “missed events” problem. Rosales and colleagues were the first to propose a method that directly uses the discrete measurements and thus does not require further idealisation of the data [65, 64]. Their algorithm estimates a discrete-time Markov model which describes the transition probabilities between states during a sampling interval rather than the so-called infinitesimal generator . Gin et al. [36] were the first to propose a method for estimating  from discretely-sampled data, their method was extended to models with arbitrary numbers of open and closed states by Siekmann et al. [74] and Siekmann et al. [70]. The current version of the software implementation of this method is available on github: https://github.com/merlinthemagician/ahmm.git. For an overview of various approaches to statistical modelling based on single-channel data, see Gin et al. [38].

The crucial advantage of MCMC methods over MLE approaches is that uncertainties can be comprehensively understood by analysing the posterior . Already marginal distributions for individual rate constants (Figure 3) are helpful for localising and quantifying uncertainties within a model .

(a)
(b)
Figure 3. Two examples for marginal distributions of rate constants. (a) shows a histogram with a well-defined mean  and a low standard deviation  which indicates a low level of parameter uncertainty whereas the histogram in (b) shows a complex multi-modal distribution which shows that multiple values of the rate constants are capable of representing the data.

But even more can be gained by analysing statistical relationships between combinations of model parameters as, for example, demonstrated by Siekmann et al. [70]. An important drawback of aggregated Markov models is non-identifiability i.e. model structures whose parameters cannot be inferred unambiguously from experimental data. Unfortunately, non-identifiable aggregated Markov models have not been completely classified [32, 33, 11]. But non-identifiability can at least be detected by analysing the posterior distribution  [70]. Thus, MCMC allows us to disentangle different causes of model uncertainty because it enables us to distinguish between parameter uncertainties due to insufficient or noisy data from pathologies in the structure of the model itself.

3.6. The Ullah et al. model

A common approach for selecting a model structure for an ion channel model (which goes back at least to the classical model by De Young and Keizer [25]) is to identify the states of the Markov model with different chemical states of the channel protein. As explained in Section 3.1, the IPR has various binding sites that allow specific ligands such as Ca and IP to bind to the channel protein and induce conformational changes of its three-dimensional structure. To account for this, model states are distinguished by how many particles of each ligand are bound to the channel. This assumption not only determines the state set of the model but also the possible transitions between states—in each state we can either bind a ligand to a free binding site or remove a ligand from an occupied binding site. The dynamics of binding and unbinding of ligands is modelled by the law of mass action so that, in principle, the model is completely specified by the number of binding sites for each ligand. However, in practice, such a model would be heavily overparameterised when fitted to experimental data, so it is necessary to simplify the model.

To illustrate this with an example, consider the recent model by Ullah et al. [84] which is representative for this approach. The model states in Figure (a)a are arranged in a grid whose horizontal axis shows how many Ca molecules are bound to the channel and whose vertical axis indicates how many IP binding sites are occupied. Thus, the position within the grid reflects for a specific model state how many Ca ions and how many IP molecules, respectively, are bound to the channel. For example, neither Ca nor IP are bound to the state C in the lower left corner whereas two Ca and four IP binding sites are occupied for the states C, O, C and O. This is also indicated by the subscript indices—the first digit stands for the number of Ca ions whereas the second digit accounts for the number of IP molecules bound to the channel. Figure (a)a shows that of the 20 possible combinations of occupying Ca, ATP and IP binding sites only a subset of eight appears in the model. This considerable reduction is due to the removal of “low occupancy states”—Ullah et al. [83] developed a perturbation theory approach that allows them to omit states with low stationary probabilities while at the same time accounting for the delays caused by passing through these states.

The model is constructed in an iterative four step process integrating several sources of data. In a first step, Ullah et al. [84] use Ca and IP dependency of the average open probability  in order to determine a minimal set of model states. By optimising an Akaike information criterion (AIC) score function, a model with five closed, C, C, C, C and C, and one open state, O, was selected as the best fit for the  data.

In a second step, the ligand-dependent average probabilities , and of being in modes characterised by three different levels of activity as well as the open probabilities in each mode (, and ) are used for assigning each of the six model states with a mode. At this step, some additional states are added because, for example, the state C must exist both in the low (C) as well as the intermediate mode (C) in order to get a good fit to the data. To account appropriately for the Ca dependency of , the open probability in the intermediate mode, an additional state O had to be introduced.

In the first two steps, Ullah et al. [84] use stationary probabilities in order to determine which states should appear in the model without considering transitions between states. In step 3 the authors infer the transitions that are needed to account for the average sojourn times , and  in the three modes whereas in step 4, data on the IPR response to rapid changes in Ca and IP (latencies) is used for determining the remaining transitions. Two additional states, C and C are introduced in order to account for the latency data.

Until this point, data is only used for determining the model structure but not for parameter estimation. The model is finally parameterised using the latency data from Mak et al. [54] or a combination of these data and single-channel time series obtained at three different constant Ca concentrations.

3.7. Siekmann et al. “Park-Drive” model

The main aims of the modelling study by Siekmann et al. [73] were first to account for switching between an inactive “park” and an active “drive” mode observed in the data set by Wagner and Yule [86]. As mentioned by Mak et al. [54] and Foskett and Mak [30], Mak and Foskett [53], the importance of modal gating is well-recognised and the implications for not appropriately capturing the timescale separation of fast opening and closing and slower switching between different activity levels is obviously unsatisfactory from a modelling point of view.

Second, these data provided the possibility to build a model of two different mammalian isoforms of the IPR, type I and type II IPR. In addition to a comparative study of type I and type II IPR, these data also include ligand-dependency of ATP in addition to IP and Ca.

Third, Siekmann et al. [73] followed a primarily statistical approach to inference, rather than deriving the model from a binding scheme as the model by Ullah et al. [84] discussed above. Based on the experience of the earlier study by Gin et al. [37] where similar data could be fitted satisfactorily by a model with four states and only one ligand-dependent pair of rate constants, the number of parameters required to account for binding of IP, Ca and ATP were likely to lead to a highly overparameterised model.

Due to these considerations, Siekmann et al. [73] made the inactive “park” and the active “drive” mode the construction principle of their model. In a first step, Markov models representing the stochastic dynamics for these two modes were constructed based on representative segments of the time series data that were characteristic for one of the two modes. Models with different numbers of states and model structures were fitted to these segments using the method by Siekmann et al. [74], Siekmann et al. [70]. It was observed that the best fits for either of the two modes across all combinations of ligands available in the large data set by Wagner and Yule [86] were quantitatively similar. This strongly suggested (consistent with Ionescu et al. [44]) that the dynamics within park and drive modes are ligand-independent and that ligand-dependent regulation of IPR activity is achieved by varying the prevalence of park or drive mode.

In a second step after both park and drive mode had been modelled separately, a model of the ligand-dependent switching between the ligand-independent modes was constructed. The structure for the full Park-Drive model (Figure (b)b) was found by connecting the Markov models of park and drive mode obtained previously with a pair of transition rates. Due to the infrequent switching between park and drive mode observed in the data it was decided that adding more than a single pair of transition rates was statistically unwarranted. The full Park-Drive model was then fitted to time series for all combinations of ligands of the study by Wagner and Yule [86]. The results of these fits established the ligand-dependency of modal gating by the IP-, Ca- and ATP-dependent variation of the two transition rates.

Probably the most important result of this study is that only models that take into account modal gating are able to accurately capture IPR kinetics. A channel whose kinetics is restricted to an active and an inactive mode can produce intermediate activity only by switching between both modes. Thus, a model that is not capable of switching between different levels of activity is misleading because it produces a constant open probability instead of alternating between highly different open probabilities. However, Cao et al. [13] showed that accounting for modal gating alone was insufficient for modelling stochastic Ca release events (puffs) that arise from the interactions of a few IPR channels. This study showed that the Park-Drive model has to be augmented by latency data [54] in order to account for the delayed response of individual channels to changes in ligand concentrations.

Constructing the Park-Drive model based on the two modes proved very useful in the study by Cao et al. [14]. The authors iteratively reduced the Park-Drive model to a two-state model that only approximates the dynamics of opening and closing within the modes and focuses on the level of activity determined by the relative prevalence of the modes. This further emphasises that switching between park and drive mode rather than stochastic dynamics within the modes is the most important mechanism of IPR regulation.

3.8. Comparison of type I and type II IPR

The experimental study by Wagner and Yule [86] not only investigated the IPR under a wide range of ligand conditions but also contrasted the behaviour of type I and type II IPR. In the models for type I and type II IPR constructed by Siekmann et al. [73] at a first glance the similarities between both subtypes are probably more obvious than the differences. First of all, it is striking that both IPR subtypes can not only be represented in the same model structure but that active and inactive modes in both channels are nearly identical. This indicates that both subtypes have the same modes and that their differences are entirely due to differences in modal gating.

One difference is that type II IPR responds more sensitively to IP, in contrast to type I IPR. The most important differences between both subtypes was found to be ATP regulation, see Wagner and Yule [86], Siekmann et al. [73] for details.

4. Using data-driven IPR models in calcium dynamics

So far we have focused on the dynamics of individual IPRs. In order to investigate the role of IPRs in calcium dynamics we will now consider the interaction of IPRs within a cluster.

4.1. Modeling calcium puffs using the Park-Drive IPR model

There is a large literature on stochastic models of calcium puffs for which we refer to the recent review by Rüdiger [66]. Here we present a simple model based on the Park-Drive model [73] which is based on the following assumptions:

  • The ER contains sufficiently high  to keep a nearly constant Ca release rate through a cluster of IP[85]. Thus, ER  dynamics is not explicitly modeled.

  • Ca fluxes through the cell membrane have little effect on the very localised Ca puffs far from cell membrane.

  • We compartmentalise our model to capture heterogeneity within a cluster of IPRs. We assume that sufficiently far away from individual channels we have a homogeneous basal Ca concentration  that slowly responds to the total Ca flux  through all IPR channels. In the vicinity of an open IPR channel this basal concentration  is elevated by a constant ; once the channel closes it instantaneously equilibrates to the basal concentration .

Furthermore, Ca buffers are not considered except a Ca fluorescence dye. With these assumptions, the model is given as follows,

(10)
(11)

where models the flux (mainly via diffusion and SERCA) removing Ca from the puff site. represents Ca leak current from the ER for stabilising the resting  of (a typical value). and represent the total dye buffer concentration and Ca-bound dye buffer concentration respectively, and the buffering process follows the mass action kinetics. is the Ca flux through open IPR, which is modeled by the production of a constant release flux rate () and number of open IPR channels (), i.e. . Each open IPR will equally contribute to the elevation of cluster , . Note that the actual  modulating each IPR is either (when it is in closed states) or (when it is in open states). Parameters values are , , , , , , and [13]. The cluster is assumed to contain 10 IPR channels.

The Park-Drive IPR model is used to simulate IPR state and coupled to the deterministic equations via a hybrid-Gillespie method [67]. However, the puff model based on the Park-Drive model fails to reproduce nonexponential interpuff interval (IPI) distribution due to the sole use of stationary single channel data (i.e. Ca is fixed during measurement) in IPR model construction. This does not allow the model to capture the transient single channel behaviour when Ca experiences a rapid change [54, 13]. Thus, the Park-Drive model is modified by incorporating time-dependent inter-mode transitions so that the transient single channel behaviour is captured. In detail, the transition rates and are changed from constants to functions of four newly-introduced gating variables,

(12)
(13)

where , , and are gating variables obeying

(14)

is the steady state which is a function of channel-sensed Ca and IP concentrations and is determined by stationary single channel data (i.e. the Park-Drive model). is the rate at which the steady state is approached. This is based on the fact that a IPR channel cannot immediately reach its steady state upon a transient change in Ca concentration [54]. The values of for , and are found to be large so that the three gating variables could be approximated by their steady states i.e. , a method called quasi-steady-state approximation. However, at low  should be very small, reflecting a very slow recovery of IPR from high Ca inhibition [54]. Note that when is sufficiently large, quasi-steady-state approximation applies and the modified IPR model reduces to the original Park-Drive model. Details about the functions and parameters can be seen in [13].

Figure 4. A simulation result of calcium puffs. represents the ratio of to its resting value. IP concentration is . Adopted from [13].

An example of simulation results using the modified Park-Drive model is give in Fig. 4. The waiting time between two successive puffs (or interpuff interval, IPI) is a key statistics to quantify the underlying process governing the emergence of puffs. Fig. 5 shows that, as at low  increases, the IPI distribution changes from nonexponential to exponential, demonstrating that the missing slow time scale in the original Park-Drive model is very crucial to explain the inhomogeneous Poisson process governing puff emergence found by (Thurley et al. [81]). The IPI distributions were generated by fitting the probability density function proposed by Thurley et al. [81] to the simulated IPI histograms [13]. The proposed IPI distribution is

(15)

where represent IPI. is the puff rate, a measure of the typical IPI (similar to average puff frequency), and is the recovery rate.

Figure 5. Dependence of IPI distribution on (indicated in the legend) at low . IP concentration is . Adopted from [13].
Figure 6. A simulation result of calcium puffs using the Ullah IPR model. IP concentration is . y-axis values indicate the number of IPR channels in corresponding states. Parameter values for the puff model remain the same.

Hence, this example shows the particular importance of considering both stationary and nonstationary data when constructing an IPR model. However, even if a model is constructed based on both data sets, it could also fail to reproduce Ca puffs. One example is the Ullah model [84] as introduced in Section 3.6. A model simulation using the same puff model (10), (11) with the Ullah model is given in Fig. 6 where the Ca signal behaves very irregularly and no puffs are clearly detected.

4.2. The role of modal gating of IPR in modulating calcium signals

The Park-Drive model (and its modified version) has the feature that IPR exist in two different modes, each of which contains multiple states, some open, some closed. Intermode transitions are important for modulating Ca signals because of their ligand- and time-dependent property. However, structure within each mode may also have substantial contribution to the formation of different Ca signals. Here, we examine the relative importance of intermode and intramode transitions using model reduction methods. By reducing the 6-state IPR model to a 2-state open/closed model, we will remove the intramodal structure, and a direct comparison between the statistics generated by the two IPR models will show the importance of intramodal structure.

The model reduction takes the following steps:

  • The low probabilities of , and (sum of which is less than 0.03 for any ) means that the IPR either rarely visit those states or have very short dwell time in those states. This allows to completely remove the three states from the 6-state model.

  • Transitions and are far larger (about two orders of magnitude) than and . By taking a quasi-steady state approximation to the transition between and , we have . Combining and to be a new state , i.e. , the 6-state model becomes a 2-state model, where represents a partially open state with Ca flux through the channel decreased by a factor of . Moreover, needs to be rescaled by due to the quasi-steady state approximation so that the effective closing rate is .

Fig. 7 shows the distributions of interpuff interval, puff duration and amplitude generated by using the 6-state IPR model (the Park-Drive model) and the reduced 2-state model. Reducing the intramodal structure does not qualitatively change the distributions but may lead to quantitative difference, which could be caused by missing open state that significantly contributes to the fluctuations of basal level of . However, if the IPR channel is not very sensitive to small fluctuations of basal , the quantitative difference is significantly reduced [14]. Thus, the fundamental process governing the generation of Ca puffs and oscillations is primarily controlled by the modal structure but not the intramodal structure which improves the model fitting to the single-channel data.

(a)
(b)
(c)
Figure 7. Comparison of interpuff interval, puff duration and amplitude between the 6-state IPR model (the Park-Drive model) and the reduced 2-state model. 199 samples for each model were used to generate A and 200 samples for B and C. Interpuff interval distributions were fit by using Eq.15 proposed by Thurley et al. [81]. Puff amplitude distributions were fit by normal distribution.

5. Conclusions

The IPR plays a major role in CICR. For this reason, more and more aspects of its behaviour have been investigated by experiments. It usually turned out that new types of data had to be explicitly included in a model to account for them. For example, in early models such as the de Young-Keizer model [25], the rate constants were determined by fitting to the  observed at different calcium concentrations. But it soon became obvious that models parameterised with  data could not be used for extrapolating the channel kinetics, i.e. the stochastic opening and closing. See Sneyd and Falcke [77] or Ullah et al. [84] for a more detailed explanation why it is impossible to infer kinetics from the ligand dependency of the open probability .

Just as kinetics cannot be inferred from  it turned out that the response of the IPR to varying ligand concentrations cannot be predicted from data collected at constant ligand concentrations. This was demonstrated by the next generation of models that were directly fitted to single-channel data, taking into account the stochastic process of opening and closing. The simplest assumption for integrating models for different ligand concentration is that the IPR adjusts instantaneously. If this was true we could represent the channel kinetics appropriately by (to give a concrete example) simply replacing the model for the kinetics at with the model for the kinetics at calcium as we increase the calcium concentration. But Cao et al. [13] showed that only after taking into account rapid-perfusion data generated by Mak et al. [54] was the model of Siekmann et al. [73] capable of generating the correct puff distribution.

It is important to note that taking into account more data does not necessarily have to lead to more complicated models. Instead, after taking into account that the simpler kinetics of modal gating should capture the part of the channel dynamics that is most important for the functional role of the IPR in CICR, Cao et al. [14] were able to reduce the six-state model by Siekmann et al. [73] to a two-state model. Thus, after interpreting experimental data in the right way, we are able to build models for the functional role of IPR that are nearly as simple as the early phenomenological models.

6. Future work

After reviewing the current state of data-driven approaches to investigating the IPR we would like to take a look at promising future directions. In order to address the particular importance of modal gating, Siekmann et al. [71] develop a novel hierarchical model structure that enables us to combine Markov models that represent the stochastic switching between modes with models that account for the characteristic opening and closing within different modes. Thus, models for both processes can be fitted separately (e.g. using the method by Siekmann et al. [74], Siekmann et al. [70]) after analysing the data with statistical method presented by Siekmann et al. [72]. This allows us to build models for modal gating following a completely data-driven approach.

More generally, we have compared two current models as representative examples for different modelling approaches, the Ullah et al. [84] and the Park-Drive model [73, 13, 14]. Although both approaches ultimately meet in the middle, their different construction principles impose different requirements for future progress. From a statistical point of view, representation of ligand interactions with a channel by mass action kinetics as in Ullah et al. [84] defines a sufficiently large search space of models. It is crucial to select from this search space an appropriately simplified model that is obtained by removing states of the full model in a consistent way. A method for model reduction is provided by Ullah et al. [83] and Ullah et al. [84] demonstrate how data can be used to statistically select from all possible simplified models. A central principle of the biophysical approach is to design models in a way that closely follows physical principles. In this context, the bond-graph approach to modelling ion channels by Gawthrop and Crampin [35], Gawthrop et al. [34] is highly relevant because it ensures that physical principles are enforced when choosing a model structure.

For models that primarily focus on a statistically satisfying representation in a first instance, the model selection problem arises again but in the other direction. Rather than starting from a model structure determined by an underlying mass action model, Gin et al. [37] and Siekmann et al. [73] iteratively increased the number of states in their model structure until further increasing the number of parameters appears statistically unwarranted. This process is time-consuming and may be computationally prohibitive if models exceed a certain number of states. Developing a method that is able to automatically compare models with an increasing number of states has proven to be difficult indicated by the few number of studies that have appeared on this subject after an early article on comparison of a finite number of models [41]. A promising new direction is the non-parametric Bayesian method developed by Hines et al. [39] which allows the authors to estimate the number of states within an ion-channel data set. Determining the required number of open and closed states in a first step may increase efficiency because it restricts the class of models which have to be compared in a second step.

Acknowledgements

Funding from NIH grant R01-DE19245 is gratefully acknowledged.

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