Electrical Control of Gastrointestinal Motility

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Chapter 3 Previous Models of ICC Activity

Portions of this chapter were published in Front. Comput. Physiol. Med. (2011), 2:29.

As experimental data on ICC electrophysiology continues to accumulate, it is imperative to coherently integrate this knowledge in order to form, test and extend hypotheses on whole cell function. Biophysically-based models provide an ideal method to achieve this integration.
Mathematical cell models can generally be classi ed as either phenomenological or biophysical (Cheng et al., 2010). Prior to the discovery that ICC act as pacemaker cells, phenomenological models were used for many years to study the electrical activity of the GI tract. These typically modelled slow waves using relaxation oscillator models in which a system of ordinary di erential equations (ODEs) produced oscillatory patterns that match the frequency and duration of slow waves, but the equations could not be directly related to an anatomical or physiological basis for slow waves (Nelsen and Becker, 1968; Publicover and Sanders, 1989; Sarna et al., 1971).
In silico studies using biophysical cell models can simulate experimental conditions by adjusting appropriate parameter values. Cell models can be used to investigate how subcellular phenomena a ect slow wave activity, and can be incorporated into multiscale models to study slow wave behaviour at the tissue, organ, and body scales. Potential applications for such simulations include evaluating physical variables not easily assessed experimentally, quantitatively relating structure with function, and potentially reducing experimental animal usage (e.g., Du et al., 2010a,c; Poh et al., 2012).

Phenomenological Cell Models

Phenomenological models had a signi cant impact on early theories of slow wave propaga-tion and entrainment (Publicover and Sanders, 1989). Nelsen and Becker (1968) published one of the earliest simulations of slow wave activity of the small intestine, consisting of a series of coupled relaxation oscillators. This model applied a generalised version of the van der Pol (1926) oscillator, transformed into a system of two rst order di erential equations with a stimulus term added to one equation. The morphology and frequency of the simulated slow waves could be controlled by adjusting the parameter values to match experimental data. Sarna et al. (1971) expanded on the concept of coupled oscillators adopted by Nelsen and Becker (1968), and demonstrated entrainment of slow waves to an ‘intact frequency’ in a linear network of coupled oscillators by incorporating forward, backward and phase-shifted backward couplings. However, the applications of relaxation oscillator slow wave models were ultimately limited by the absence intracellular details, particularly ion channels. Consequently, Publicover and Sanders (1989) discussed the limitations of the relaxation oscillator based models, including their inability to represent the e ects of pharmacological agents or electrical stimuli on slow wave activity, and the mismatches between the morphologies of simulated slow wave activity and intracellular slow wave recordings. Phenomenological models have now been superseded by more sophisticated modelling methods providing a more physiologically realistic representation of slow waves (Cheng et al., 2010).

Biophysical Cell Models

Biophysical cell models are usually based on the mathematical approach developed by Hodgkin and Huxley (1952), allowing individual ion currents to be quantitatively evaluated under e ects of parameters with meaningful physical quantities, such as temperature, ion concentration, and voltage. Hodgkin-Huxley type models represent the plasma membrane lipid bilayer as a capacitor connected in parallel with variable resistors representing the ion channels in the membrane. The change in membrane potential (V_m) is dependent on the total current through the ion channels (I_ion) and membrane capacitance (C_m).
The current through each class of ion channel is governed by gating variables that describe the sensitivity and time-dependence of the ion channel to voltage and other stimuli. The gating of voltage-dependent ion channels is typically represented using the Boltzmann equation (Eq. 2.1).

Previous ICC Models

The rst model to explicitly represent ICC as a separate cell type was a phenomenological model by Aliev et al. (2000). The development of biophysical cell models of ICC is at a relatively nascent stage, with the rst full biophysical ICC model created in 2006 (Youm et al., 2006). Two further biophysical ICC models (Corrias and Buist, 2008; Faville et al., 2009) and two PMU models (Faville et al., 2008; Means and Sneyd, 2010) have been published more recently. A primary focus of biophysical ICC models is to quantitatively understand the signalling pathways that give rise to slow wave activity. The following sections present each of these models and discusses their respective merits, as well as potential areas for improvement. Note that these models sometimes use di erent terminology to refer to the same ion channels. For the sake of simplicity, this thesis will use a common symbol to refer to each type of current where it is clear that the same ion channel is being modelled, and will also provide the symbol used by the original authors.
Aliev Slow Wave Model
Aliev et al. (2000) developed the rst model of slow wave activity to incorporate separate representations of ICC and SMC. The simulated ICC and SMC were arranged in a one-dimensional line in two layers representing the ICC-MY and longitudinal muscle of the small intestine. Parameters could be adjusted to alter physiological behaviour, such as frequency of oscillation and resistivity of membrane coupling between cells. The Aliev model was nevertheless a phenomenological model based on the FitzHugh-Nagumo neuron model (FitzHugh, 1961; Nagumo et al., 1962), with dimensionless parameters that were scaled to the appropriate units.
The Aliev model was used in anatomically realistic multiscale models of GI electrical activity (Cheng et al., 2007; Lin et al., 2006; Pullan et al., 2004). However, it had become evident that a biophysically-based approach to modelling slow waves was necessary to take advantage of burgeoning discoveries about the electrophysiology of ICC and SMC.

Youm Slow Wave Model

The ICC model by Youm et al. (2006), partially based on cardiac cell models, was the rst biophysical cell model to include the ion channels and intracellular Ca²⁺ transients that were thought to contribute to slow wave activity. The model included four ion channels: an inward recti er K⁺ current (I_K1), the L-type Ca²⁺ current (I_CaL), a voltage-dependent DHP-resistant current (I_VDDR), and a Ca²⁺-activated autonomous inward current (I_AI) carried by K⁺, Ca²⁺ and Na⁺ ions. Three ion transporters were also included in the model:
NCX (I_NaCa), a Na⁺/K⁺ pump (I_NaK), and a PMCA pump (I_PMCA). Figure 3.1 depicts schematic diagram of the model showing all the ion channels and transporters. The time-dependent membrane potential is described by the following expression,  where C_m was set at 25 pF, the membrane capacitance of an ICC isolated from mouse small intestine (Koh et al., 2002; Youm et al., 2006), and I_s is a stimulus current. Figure 3.4A shows slow wave activity simulated using the Youm et al. (2006) model.
The intracellular Ca²⁺ transient in the ICC model by Youm et al. (2006) is governed by four Ca²⁺ uxes: an SR (or ER) uptake current, I_up; IP₃-mediated Ca²⁺ release from the SR, I_IPR; a di usive Ca²⁺ leak current from the SR, I_leak; and the net sum of I_CaL, I_AI, I_VDDR, I_PMCA, and I_NaCa. The metabolism of IP₃ was governed by a three-state model of IP₃, IP₄, and PIP₂ production and degradation, with voltage- and Ca²⁺-dependent rate constants. The synthesis of IP₃ controls the conductance of I_IPR. The SR was modelled with separate sites for Ca²⁺ uptake from and release to the cytoplasm. SR Ca²⁺ uptake is governed by three currents: I_up; I_leak; and I_tr, a transfer current between the uptake and release sites on the SR. Ca²⁺ release at the SR release site is governed by I_tr and I_IPR. The time-dependent intracellular Ca²⁺ transient is described by the following expression, where z_Ca denotes the valence of Ca²⁺, F is the Faraday constant, V_c denotes the cell volume (712:5 µm³), and I_AI(Ca) is the Ca²⁺ component of I_AI (Youm et al., 2006).

Corrias and Buist Slow Wave Model

The ICC model by Corrias and Buist (2008) was principally based on the NSCC Hypothesis described in Section 2.9.1. The model ICC contained a single PMU with ER, mitochondria, and a small cytosolic subspace, representing the aggregate of all the PMUs in an ICC, as shown in Figure 3.2. A Ca²⁺-inhibited NSC channel in the PMU membrane generated current, I_NSCC, in response to Ca²⁺ cycling in the PMU. Because the PMUs were represented by a single bulk PMU, the simulated slow wave plateau phase was not generated by unitary potential summation, but by whole cell current ow through several ion channels, particularly a Ca²⁺-activated Cl channel, I_Cl.
The model included nine types of ion channels and one ion transporter in the bulk cytoplasm: I_VDDR, I_CaL (called I_L-type in the original literature), the delayed-recti er K_V1.1 current (I_Kv1.1, originally called I_kv11), the ERG K⁺ current (I_ERG), a Ca²⁺-activated K⁺ conductance (I_BK), a background K⁺ leak current (I_K(B), originally called I_Kb), the voltage-dependent Na⁺ current (I_Nav1.5, originally called I_Na), I_NSCC, I_Cl, and a Ca²⁺ extrusion mechanism representing the action of PMCA and NCX transporters (J_Ca(Ext), called I_Ca-EXT in the original literature). Figure 3.4B shows slow wave activity simulated using the Corrias and Buist ICC model. The time-dependent membrane potential is described by the following expression,
C_m ^dVm = I_VDDR +I_CaL +I_Kv1.1 +I_ERG +I_BK +I_K(B) +I_Nav1.5 +I_NSCC +I_Cl +I_Ca(Ext): (3.4) dt
Corrias and Buist (2008) adapted an extensive description of intracellular Ca²⁺ dynamics from Fall and Keizer (2001) to represent Ca²⁺ handling in the PMU. Ca²⁺ ux between the PMU cytosolic subspace and the greater cytoplasm is governed by a passive di usive current (J_leak). The Ca²⁺ handling dynamics of the mitochondria and ER induce Ca²⁺ oscillations in the cytosolic subspace, from which Ca²⁺ di uses into the bulk cytoplasm through J_leak. This leads to a global increase in [Ca²⁺]_i, which in turn activates the Ca²⁺-dependent ion conductances in the cell model: I_CaL, I_BK, and I_Cl. Conversely, I_NSCC is activated by the falling phase of the Ca²⁺ oscillations within the PMU. The time-dependent intracellular Ca²⁺ transient is described by the following expression, where f_c denotes the cytosolic free Ca²⁺ proportion (set to 0.01), F denotes the Faraday constant, and V_c is the cytosolic volume (700 µm³).

Contents
Abstract 
Acknowledgements 
Contents 
List of Figures 
List of Tables 
Glossary 
1 Introduction 
1.1 Motivation and Aims
2 Gastrointestinal Electrophysiology 
2.1 Electrical Control of Gastrointestinal Motility
2.2 Interstitial Cells of Cajal
2.3 Methods for Investigating Gastrointestinal Electrical Activity
2.4 Calcium Oscillations in ICC
2.5 Store-Operated Calcium Entry
2.6 Ion Channels Found in ICC
2.7 Unitary Potentials
2.8 Voltage-Dependent Entrainment
2.9 Pacemaker Hypotheses
3 Previous Models of ICC Activity 59
3.1 Phenomenological Cell Models
3.2 Biophysical Cell Models
3.3 Previous ICC Models
3.4 The Need for a New Model
4 Sodium Channel Mechanosensitivity 
4.1 Mechanosensitivity of Ion Channels
4.2 The Roles of NaV1.5 in ICC and SMC
4.3 Sodium Channel Model
4.4 Verifying the Role of NaV1.5 in the ICC Model
4.5 Modelling Stretch in ICC
4.6 Future Directions
5 Ano1 Channel Model 
5.1 The Ano1 Model
5.2 Results
5.3 Discussion
6 ICC Model 
6.1 Model Development
6.2 Results
6.3 Discussion
7 Conclusions 
7.1 The ICC Pacemaker Mechanism
7.2 Modelling the Ano1 Channel
7.3 Modelling Slow Waves in ICC
7.4 Modelling ICC Mechanosensitivity
7.5 Concluding Remarks
References
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Keeping Pace with Interstitial Cells of Cajal: Modelling Gastrointestinal Electrophysiology