Transcript Slide 1
Mathematical Modelling of Phage Dynamics: Applications in STEC studies Tom Evans 1. Some biology 2. Molecular level modelling 3. Population level modelling 1. Some biology • Shiga toxin-encoding bacteriophages (Stx phages). • Encode the toxins Stx1 or Stx2 • Temperate, i.e. following infection of an E. coli cell by Stx phage, either lysis or lysogeny will occur • Lysis leads to the death of the bacterial cell • If the lysogenic pathway is selected, the phage inserts its DNA into the bacterial chromosome. Thus the bacteria survives and gains some new genes. • ST strains of E. coli can cause disease in humans • They enter the human food chain via livestock • Symptoms include bloody diarrhoea and kidney failure • What are the processes at the molecular level which govern the lysis/lysogeny switching mechanism and hence the rate at which toxins are released? • Can a resident population of phages (bacteria) be invaded by a mutant strain of phages (bacteria)? For example, can a more virulent strain of phages invade a less virulent strain, and if so what are the implications for the spread of toxins? 2. Molecular level modelling • Suppose that a single E. coli cell is infected by a single Stx phage. • Then either the lytic or the lysogenic response will be induced. • What determines which of these outcomes occurs? • Immediately following infection, various genes are expressed, resulting in the production of proteins including Cro and CI • The levels at which the genes cro and cI are expressed determine whether lysis or lysogeny occurs. • A cell becomes committed to lysogeny if the concentration of CI is greater than the concentration of Cro at the end of the 35-minute cell cycle. • A model can be used to simulate the lysis/lysogeny decision making process. • Arkin et al (1998) describes a model of the system, and uses a “stochastic” computer algorithm to simulate its behaviour. • The model consists of sets of: – Chemical species – Chemical reactions between the species – The rates at which the reactions take place • The model allows for random behaviour • Thus, the rate constants are actually probabilities per unit time of a reaction event taking place. • Low numbers of molecules mean that random effects are likely to be significant • For example, in order for the CI protein to function it must first form a dimer. • Therefore the model must include a reaction which represent dimerization of CI: CI + CI CI2 • We must also specify a rate law, which determines how frequently this reaction occurs: k1*[CI][CI] – k2*[CI2] • All the important reactions and their rate laws must be included in the model. Complicated. • The model can then be used to simulate the levels of the Cro and CI proteins during the 35-minute cell cycle. • The model must be run many times to establish what happens on average. 35 0 Time (minutes) Summary of Molecular Level Modelling • A version of the Arkin model is available as part of the “Dynetica” simulation tool. • Can be used to run simulations of the lysis-lysogeny switching mechanism for Stx phages. • No new work as yet. 3. Population Dynamics and Evolution • As well as considering what happens at the molecular level, we can look at how populations of phages and bacteria interact. • Assume that populations of bacteria, phages and lysogens exist in a given environment. • Also assume that resources flow into this environment • We can write down equations which specify the rates at which the populations change over time. • Stewart and Levin (1983) and Mittler (1996). • The equations include factors such as – The adsorption rate (delta) – The phage burst size (beta) – The proportion of phage infections which lead to lysogeny (lambda) – The ability of bacteria to take in resources from the environment (psi) Population Dynamics Equations • Once the equations have been specified, a computer program can be used to simulate the populations over time. • These simulations can be used to help answer questions about the way in which phage and bacterial populations change over time. • As well as looking at the population dynamics of phages and bacteria, we can look at how different characteristics evolve through successive generations • For example, suppose that E. coli strains differ only in two parameters: (i) their resistance to phage infection and (ii) their ability to take in resources from their environment • Also suppose that we have a resident population which consists of a single strain of E. coli (i.e. all members of this population have the same values for the two parameters) • If a few mutant E. coli cells with a different values of one or both parameters emerges, will this mutant population be able to grow in number, or will it just die out? • The answer depends on the relative fitness levels of the resident and mutant populations. A Fitness Equation • 1=resident • 2=invading mutant • Fitness of invading mutant • Where A Pairwise Invasibility Plot (PIP) Summary of Population Level Modelling • Used to model population dynamics and evolution • Simulate populations of phages and bacteria. Find equilibria. • Establish “optimal” evolutionary strategies for phages and bacteria • New work Conclusion • Modelling at two different levels: Molecular and Population • Ultimately the two levels should be combined in a single model • Not easy to do