No abstract

Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461-3470, 1987; Clarke and Jiang in J Chem Phys 99:4464-4476, 1993; Gatermann et al. in J Symb Comput 40:1361-1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks.

Based on the description of an outbreak of foot-and-mouth disease (FMD), a particle model is developed describing the most important properties of this epidemic. Also control measures (mass and ring vaccination) are implemented. This model shows the expected behavior in simulations. Since it is impossible to treat this model analytically, we use ideas of branching processes on two levels to derive a caricature of the particle model. In simulations it is shown that this caricature exhibits similar behavior as the particle system. It is possible to analyze the caricature and, in this way, to obtain expressions for the most important quantities like the reproduction number or the expected final number of infected individuals etc. In this way mass vaccination and ring vaccination can be compared and control strategies can be optimized.

SummaryCyanobacteria possess a differentiated membrane system and transport proteins into both the periplasm and thylakoid lumen. We have used green fluorescent protein (GFP)-tagged constructs to study the Tat protein transporter and Rieske Tat substrates in Synechocystis PCC6803. The Tat system has been shown to operate in the plasma membrane; we show here that it is also relatively abundant in the thylakoid membrane network, indicating that newly synthesized Tat substrates are targeted to both membrane systems. Synechocystis contains three Rieske ironsulphur proteins, all of which contain typical twinarginine signal-like sequences at their N-termini. We show that two of these proteins (PetC1 and PetC2) are obligate Tat substrates when expressed in Escherichia coli. The Rieske proteins exhibit differential localization in Synechocystis 6803; PetC1 and PetC2 are located in the thylakoid membrane, while PetC3 is primarily targeted to the plasma membrane. The combined data show that Tat substrates are directed with high precision to both membrane systems in this cyanobacterium, raising the question of how, and when, intracellular sorting to the correct membrane is achieved.