Stephen Baigent

P Matrix Properties, Injectivity, and Stability in Chemical Reaction Systems

In this paper we examine matrices which arise naturally as Jacobians in chemical dynamics. We are particularly interested in when these Jacobians are P matrices (up to a sign change), ensuring certain bounds on their eigenvalues, precluding certain behavior such as multiple equilibria, and sometimes implying stability. We first explore reaction systems and derive results which provide a deep connection between system structure and the P matrix property. We then examine a class of systems consisting of reactions coupled to an external rate-dependent negative feedback process and characterize conditions which ensure that the P matrix property survives the negative feedback. The techniques presented are applied to examples published in the mathematical and biological literature.

Cells coupled by voltage-dependent gap junctions: the asymptotic dynamical limit

We study the steady state and dynamical properties of a pair of cells coupled by a voltage-dependent gap junction. The cells have linear membrane properties, and the gap junction is modelled using a simple Markov chain with a voltage-dependent transition matrix. We first show that the voltage-independent case is globally convergent using energy dissipation as a Lyapunov function for the cells, and standard results on the convergence of homogeneous Markov chains for the junction. For the voltage-dependent case, we use the difference in cell and gap junction time scales to reduce the coupled equations for cells and the gap junction to a single equation for the gap junction, but with a transition matrix that depends upon the current gap junction state. We identify cooperativity as key property behind the global convergence of Markov chains and investigate convergence of the voltage-dependent system by establishing some conditions under which cooperativity is preserved.

Arterial ammonia levels in cirrhosis are determined by systemic and hepatic hemodynamics, and by organ function: a quantitative modelling study

Hyperammonaemia is a common complication of chronic liver failure. Two main factors are thought to underlie this complication: a loss of hepatic detoxification function and the development of portosystemic shunting. However, few studies have tried to quantify the importance of portosystemic shunting. Here, we used a theoretical approach to test the hypothesis that the development of portosystemic shunting is sufficient to cause hyperammonaemia in cirrhosis.

Geometry of carrying simplices of 3-species competitive Lotka?Volterra systems

We investigate the existence, uniqueness and Gaussian curvature of the invariant carrying simplices of 3 species autonomous totally competitive Lotka–Volterra systems. Explicit examples are given where the carrying simplex is convex or concave, but also where the curvature is not single-signed. Our method monitors the curvature of an evolving surface that converges uniformly to the carrying simplex, and generally relies on establishing that the Gaussian image of the evolving surface is confined to an invariant cone. We also discuss the relationship between the curvature of the carrying simplex near an interior fixed point and its Split Lyapunov stability. Finally we comment on extensions to general Lotka–Volterra systems that are not competitive.

Fixed point global attractors and repellors in competitive Lotka–Volterra systems

Zeeman and Zeeman [E.C. Zeeman and M.L. Zeeman, From local to global behavior in competitive Lotka–Volterra systems, Trans. Amer. Math. Soc. 355 (2003), pp. 713–734] show that if a strongly competitive Lotka–Volterra system (i) has a unique interior fixed point p and (ii) the carrying simplex Σ lies below (above) the strongly balanced tangent plane to Σ at p then the system has no periodic orbits and p is a global attractor (repellor) relative to Σ. Condition (ii) is then translated into the definiteness of a certain quadratic function on the tangent plane, which is equivalent to the definiteness of an (N − 1) × (N − 1) real symmetric matrix that can be computed. Here we adapt these methods to show that the above conclusions are still true without the assumption (i). Hence, our results apply to globally attracting or repelling fixed points on the boundary, as well as in the interior, of . Moreover, the algebraic condition for global attraction also implies global asymptotic stability of the fixed point. We also show that the global attraction holds not just relative to Σ, but also relative to the interior of the first quadrant.

Convexity-preserving flows of totally competitive planar Lotka-Volterra equations and the geometry of the carrying simplex

We show that the flow generated by the totally competitive planar Lotka-Volterra equations deforms the line connecting the two axial equilibria into convex or concave curves, and that these curves remain convex or concave for all subsequent time. We apply the observation to provide an alternative proof to that given by Tineo in 2001 that the carrying simplex, the globally attracting invariant manifold that joins the axial equilibria, is either convex, concave or a straight-line segment.

Convexity of the carrying simplex for discrete-time planar competitive Kolmogorov systems

We consider the geometry of carrying simplices of discrete-time competitive Kolmogorov systems. An existence theorem for the carrying simplex based upon the Hadamard graph transform is developed, and conditions for when the transform yields a sequence of convex or concave graphs are determined. As an application it is shown that the planar Leslie–Gower model has a carrying simplex that is convex or concave.

Monotone dynamics of two cells dynamically coupled by a voltage-dependent gap junction.

We introduce and analyse a simple model for two non-excitable cells that are dynamically coupled by a gap junction, a plaque of aqueous channels that electrically couple the cells. The gap junction channels have a low and high conductance state, and the transition rates between these states are voltage-dependent. We show that the number and stability of steady states of the system has a simple relationship with the determinant of the Jacobian matrix. For the case that channel opening rates decrease with increasing trans-junctional voltage, and closing rates increase with increasing trans-junctional voltage, we show that the system is monotone, with tridiagonal Jacobian matrix, and hence every initial condition evolves to a steady state, but that there may be multiple steady states.

Integrating Biosystem Models Using Waveform Relaxation

Modelling in systems biology often involves the integration of component models into larger composite models. How to do this systematically and efficiently is a significant challenge: coupling of components can be unidirectional or bidirectional, and of variable strengths. We adapt the waveform relaxation (WR) method for parallel computation of ODEs as a general methodology for computing systems of linked submodels. Four test cases are presented: (i) a cascade of unidirectionally and bidirectionally coupled harmonic oscillators, (ii) deterministic and stochastic simulations of calcium oscillations, (iii) single cell calcium oscillations showing complex behaviour such as periodic and chaotic bursting, and (iv) a multicellular calcium model for a cell plate of hepatocytes. We conclude that WR provides a flexible means to deal with multitime-scale computation and model heterogeneity. Global solutions over time can be captured independently of the solution techniques for the individual components, which may be distributed in different computing environments.

Some aspects of gap junction dynamics in embryonic systems

Effective intercellular communication is essential for the proper integration of any multicellular system into a functioning syncytium. In many tissues an intercellular link is provided by arrays of aqueous protein channels known as gap junctions. Individual cells communicate with their neighbours by the exchange of ions and small molecules through gap junctions. In this way a biological signal may be relayed from one cell to a distant neighbour via a chain of cells.