Lymphatic system defects are involved in a wide range of diseases, including obesity, cardiovascular disease, and neurological disorders, such as Alzheimer’s disease. Fluid return through the lymphatic vascular system is primarily provided by contractions of muscle cells in the walls of lymphatic vessels, which are in turn driven by electrochemical oscillations that cause rhythmic action potentials and associated surges in intracellular calcium ion concentration. There is an incomplete understanding of the mechanisms involved in these repeated events, restricting the development of pharmacological treatments for dysfunction. Previously, we proposed a model where autonomous oscillations in the membrane potential (M-clock) drove passive oscillations in the calcium concentration (C-clock). In this paper, to model more accurately what is known about the underlying physiology, we extend this model to the case where the M-clock and the C-clock oscillators are both active but coupled together, thus both driving the action potentials. This extension results from modifications to the model’s description of the IP3 receptor, a key C-clock mechanism. The synchronised dual-driving clock behaviour enables the model to match IP3 receptor knock-out data, thus resolving an issue with previous models. We also use phase-plane analysis to explain the mechanisms of coupling of the dual clocks. The model has the potential to help determine mechanisms and find targets for pharmacological treatment of some causes of lymphoedema.
A dual-clock-driven model of lymphatic muscle cell pacemaking to emulate knock-out of Ano1 or IP3R
Disclosures: The authors declare no competing interests exist.
A similar degree of complexity is manifested by the smooth muscle in the wall of those small arterial blood vessels which exhibit oscillatory vasomotion on a rather rapid time scale, in addition to their traditionally recognised function of sustaining a standing level of vascular tone and thus hydraulic resistance.
In this paper, we term the calcium-concentration equivalent of the AP a calcium flash.
There are three IP3 receptor isoforms; it has not yet been ruled out that there could be a degree of compensatory expression of IP3R2 or IP3R3.
However, these ratios are not particularly meaningful since the control and IP3R-KO traces come from different animals.
The mechanisms are allocated to the clocks according to which clock they act on, or equivalently of which ODE they are part. For example, the Ano1 Cl− channel acts directly on the membrane potential (Eq. 4) and so is part of the M-clock despite receiving Ca2+ signals. The L-type Ca2+ channel acts on both clocks (Eqs. 1 and 4) and so is technically part of both.
We here follow precedent (Dupont et al., 2016) in assuming that the plasma membrane Ca2+ ATPase pump operation is stoichiometrically neutral (as many positive charges on monovalent hydrogen ions are imported as are exported on bivalent calcium ions), and thus no term in Jce appears in Eq. 4. The exact stoichiometry is uncertain (Brini and Carafoli, 2011).
In fact there is no true frequency to the Z(t) oscillation under Ano1-KO, because the underlying C-clock frequency is not synchronised with the APs, even at the apparent reduced rate of one AP per four flashes—there is continuous slippage of the AP phase relative to the latest flash. In consequence the amount to which L-type current contributes to the largest flashes varies over a longer passage of time than is shown here. In dynamical systems terms, the oscillation is quasi-periodic.
In fact, this is rather a projection onto two dimensions of the three-dimensional (Z,Y,n)-space, since the trajectory does not keep to a constant value of Y. Similarly, the two positions shown for the Z-nullcline are for different values of Y.
In their Fig. 9, Imtiaz et al. (2007) demonstrated a wider [IP3] range for oscillation in the Dupont–Goldbeter C-clock-only model upon which their model was founded, but we here refer to their entire model as reinterpreted on a more physical basis and with more realistic parameter values by Hancock et al. (2022).
Multicellular communication of APs and calcium transients could if required be modelled by adding extra terms to the equations to represent the coupling between cells, as demonstrated by Imtiaz et al. (2007).
Much more complex models for the IP3 receptor exist, but these necessitate many ODEs, taking the resulting ODE system out of the realm where even approximate phase-plane analysis is feasible or useful.
Experimentally quantifying the L-type current activation and inactivation range in LMCs would help increase the accuracy with which the model can fit the real situation.
Imtiaz et al. (2007) did not incorporate RyR channels in their model, as ryanodine had no effect on the vasomotion of their guinea pig lymphatic vessels, nor did RyR inhibitors alter spontaneous transient depolarisations (von der Weid et al., 2008). However, functional RyRs have recently been reported in rat lymphatic vessels (Jo et al., 2019; Stolarz et al., 2019). Our unpublished observations suggest that RyR channels are also expressed in mouse LMCs, but play a minimal role when IP3R1 is present.
All Na+ and K+ fluxes have been simplified to a single linear current. Furthermore, potentially significant Ca-activated (e.g., BKCa) or voltage-dependent (e.g., KV) channels are absent. This is analogous to the FitzHugh–Nagumo reduction of the four-dimensional Hodgkin–Huxley model to two dimensions, through recognition of the slow dynamics of K+ channels (FitzHugh 1961; Nagumo et al., 1962; Keener and Sneyd, 2009).
This work is part of a special issue on Structure and Function of Ion Channels in Native Cells and Macromolecular Complexes.
