Quasi-steady-state reduction in the analysis of biophysical models and excitation-contraction coupling in airway smooth muscle cells

Abstract

This thesis addresses two topics related to the mathematical modelling of biophysical systems. The rst part of the thesis investigates quasi-steady-state reduction (QSSR), which is a technique commonly used for dimension reduction for systems with multiple timescales. Many biophysical models have the property that some variables in the model evolve much faster than others. A common rst step in the analysis of such systems is to simplify the model by assuming that some of the fastest variables equilibrate instantaneously; this approach is known as QSSR. QSSR is intuitively satisfying but is not always mathematically justi ed, with problems known to arise, for instance, in examples in which the full model has oscillatory solutions. In models with oscillations, a simpli cation by QSSR may lead to a model with signi cantly di erent dynamics compared to the original model. In the rst part, we focus on the e ect of QSSR on models in which oscillatory solutions arise via one or more Hopf bifurcations. We rst illustrate the problems that can arise by applying QSSR to a selection of well-known models. We then prove that Hopf bifurcations that involve fast and slow variables (i.e., singular Hopf bifurcations) are generically preserved under QSSR so long as a fast variable is kept in the simpli ed system. Furthermore, we argue that Hopf bifurcations that involve only slow variables are not a ected by QSSR, and Hopf bifurcations that primarily involve fast variables may be eliminated by QSSR. The persistence of a Hopf bifurcation does not guarantee that the resulting periodic orbits are unchanged by QSSR. We show that the criticality of a Hopf bifurcation may be changed by QSSR, with a resultant change in the stability of periodic orbits near the Hopf bifurcation. Furthermore, we show that QSSR can a ect the amplitude and frequency of periodic orbits and introduce or remove homoclinic bifurcations. Finally, we present some guidelines for the application of QSSR if one wishes to use the method while minimising the risk of inadvertently destroying essential features of the original model. In the second part of this thesis we study the feedback between changes in voltage across the plasma membrane and the intracellular Ca2+ handling in airway smooth muscle cells (ASMC). ASMC are able to contract, and the contraction can result in pathological airway obstruction in asthma or airway hyperresponsiveness. Contraction of ASMC is induced by an increased cytosolic Ca2+ concentration ([Ca2+]i). Typically, [Ca2+]i oscillates during prolonged smooth muscle contraction and the oscillation frequency a ects the strength of contraction. Oscillations of [Ca2+]i result from either depolarization followed by increased [Ca2+]i and calcium-induced Ca2+ release from internal stores, or the binding of an agonist which produces inositol (1,4,5)-trisphosphate and leads to Ca2+ release from internal stores. Previous models of [Ca2+]i oscillations in ASMC have included the voltage across the plasma membrane as a parameter which modi es the Ca2+ in ux [19, 23, 94]. However, evidence suggests that the voltage changes dynamically during [Ca2+]i oscillations [98]. We combine mathematical models of intracellular Ca2+ receptors and plasma membrane channels and use experimental data from mouse lung slices to assess the e ect of voltage dynamics on the [Ca2+]i oscillations. Furthermore, we look at the contribution of di erent Ca2+ channels in the plasma membrane, in particular the voltage-gated and the storeoperated Ca2+ channels. The main results of this part of the thesis are: • Agonist-induced [Ca2+]i oscillations are not signi cantly a ected by changes in voltage. Variations in voltage, due to the intrinsic voltage dynamics, only lead to small modulations of the Ca2+ in uxes. • During depolarization-induced [Ca2+]i oscillations the voltage-gated Ca2+ channel contributes most to the Ca2+ in ux; during agonist-induced [Ca2+]i oscillations the store-operated Ca2+ channel contributes most to the Ca2+ in ux. • During agonist-induced [Ca2+]i oscillations the internal stores do not fully deplete. The Ca2+ concentration in internal stores plateaus near 80% of the resting concentration during agonist-induced oscillations.