Balance simplices of 3-species May-Leonard systems.

We investigate the existence of a two-dimensional invariant manifold that attracts all nonzero orbits in 3 species Lotka-Volterra systems with identical linear growth rates. This manifold, which we call the balance simplex, is the common boundary of the basin of repulsion of the origin and the basin of repulsion of infinity. The balance simplex is linked to ecological models where there is 'growth when rare' and competition for finite resources. By including alternative food sources for predators we cater for predator-prey type models. In the case that the model is competitive, the balance simplex coincides with the carrying simplex which is an unordered manifold (no two points may be ordered componentwise), but for non-competitive models the balance simplex need not be unordered. The balance simplex of our models contains all limit sets and is the graph of a piecewise analytic function over the unit probability simplex.

The balance simplex in non-competitive 2-species scaled Lotka-Volterra systems.

Explicit expressions in terms of Gaussian Hypergeometric functions are found for a 'balance' manifold that connects the non-zero steady states of a 2-species, non-competitive, scaled Lotka-Volterra system by the unique heteroclinic orbits. In this model, the parameters are the interspecific interaction coefficients which affects the form of the solution used. Similar to the carrying simplex of the competitive model, this balance simplex is the common boundary of the basin of repulsion of the origin and infinity, and is smooth except possibly at steady states.

Mathematical Model of Ammonia Handling in the Rat Renal Medulla.

The kidney is one of the main organs that produces ammonia and release it into the circulation. Under normal conditions, between 30 and 50% of the ammonia produced in the kidney is excreted in the urine, the rest being absorbed into the systemic circulation via the renal vein. In acidosis and in some pathological conditions, the proportion of urinary excretion can increase to 70% of the ammonia produced in the kidney. Mechanisms regulating the balance between urinary excretion and renal vein release are not fully understood. We developed a mathematical model that reflects current thinking about renal ammonia handling in order to investigate the role of each tubular segment and identify some of the components which might control this balance. The model treats the movements of water, sodium chloride, urea, NH3 and [Formula: see text], and non-reabsorbable solute in an idealized renal medulla of the rat at steady state. A parameter study was performed to identify the transport parameters and microenvironmental conditions that most affect the rate of urinary ammonia excretion. Our results suggest that urinary ammonia excretion is mainly determined by those parameters that affect ammonia recycling in the loops of Henle. In particular, our results suggest a critical role for interstitial pH in the outer medulla and for luminal pH along the inner medullary collecting ducts.

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

BACKGROUND & AIMS: 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. METHODS: Two mathematical models are developed. The first one describes the main vascular resistances of the circulation and is used to provide scenarios for the distributions of organ blood flow in cirrhosis, which are necessary to run the second model. The second model predicts arterial ammonia levels resulting from ammonia metabolism in gut, liver, kidney, muscle and brain, and the distribution of organ blood flow. RESULTS: The fraction of gastrointestinal blood flow shunted through collaterals was estimated to be 41% in mild cirrhosis, 69% in moderate and 85% in severe cases. In the second model, the redistribution of organ blood flow associated with severe cirrhosis was sufficient to cause hyperammonaemia, even when the hepatic detoxification function and the ammonia production were set to normal. CONCLUSIONS: The model indicates that the development of portosystemic shunting in cirrhosis is sufficient to cause hyperammonaemia. Interventions that reduce the fraction of shunting may be future targets of therapy to control severity of hyperammonaemia.

Quantifying the likelihood of co-existence for communities with asymmetric competition.

Trade-offs in performance of different ecological functions within a species are commonly offered as an explanation for co-existence in natural communities. Single trade-offs between competitive ability and other life history traits have been shown to support a large number of species, as a result of strong competitive asymmetry. We consider a single competition-fecundity trade-off in a homogeneous environment, and examine the effect of the form of asymmetry on the likelihood of species co-existing. We find conditions that allow co-existence of two species for a general competition function, and show that (1) two species can only co-exist if the competition function is sufficiently steep when the species are similar; (2) when competition is determined by a linear function, no more than two species can co-exist; (3) when the competition between two individuals is determined by a discontinuous step function, this single trade-off can support an arbitrarily large number of species. Further, we show analytically that as the degree of asymmetry in competition increases, the probability of a given number of species co-existing also increases, but note that even in the most favourable conditions, large numbers of species co-existing along a single trade-off is highly unlikely. On this basis, we suggest it is unlikely that single trade-offs are able to support high levels of bio-diversity without interacting other processes.

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.

A physiological model of cerebral blood flow control.

The construction of a computational model of the human brain circulation is described. We combine an existing model of the biophysics of the circulatory system, a basic model of brain metabolic biochemistry, and a model of the functioning of vascular smooth muscle (VSM) into a single model. This represents a first attempt to understand how the numerous different feedback pathways by which cerebral blood flow is controlled interact with each other. The present work comprises the following: Descriptions of the physiology underlying the model; general comments on the processes by which this physiology is translated into mathematics; comments on parameter setting; and some simulation results. The simulations presented are preliminary, but show qualitative agreement between model behaviour and experimental results.

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.