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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