Who is the app suitable for?
This app is best suited for geneticists interested in the evolution of
bacterial populations of phenotypes/phasotypes dictated by phase variation (PV).
Exposure to fluctuating environments is thought to drive the evolution of genes with
a binary (ON/OFF) switching property, often called phase variation, in bacteria
populations. Evolution of PV is influenced by mutation characterisitics and selection
coefficients and bottlenecks. In this app, we propose a discrete-time discrete-space stochastic
model for PV which takes into account mutation and selection mechanisms.
What are the outcomes of the app?
With a starting distribution of phasotypes,
the app uses mutation rates for each gene with advantageous selection chosen for
particular phasotypes to find the expected distribution after a given number of
estimated generations. In addition, the app also finds the expected stationary distribution,
the amount of generations required to reach this stationary distribution, and produces plots
for each of the three distributions featured. The basic features of this interactive model
are explained through a step-by-step guide.
How does the model work?
As input parameters, the model has (in order of that seen in the app) the number of genes
with the ON/OFF PV property, the initial distribution of gene states in the population,
the number of bacterial generations
n
, mutation rates per gene of OFF-to-ON and ON-to-OFF
switching probabilities per generation, and a vector of fitness parameters (if selection is
turned ON). The output of the model is the distribution of gene states after the selected
n
generations, with a stationary distribution and time to stationary distribution also recorded.
The stationary distribution as
n
goes to infinity is unique and not conditional on the initial
distribution choice.
The model is derived under the assumption of a near infinite population size. Derivation of the model is
based on an idea of splitting of mutation and selection at each time step and on averaging of all
possible realisations of the evolution of the branching tree. In terms of its properties, the model
resembles a nonlinear Markov chain. When all the fitness parameters are equal, the model degenerates to
a model that considers mutation without selection.
Why are there specific preset values on the input page?
The idea is that beginners to the app or those who wish only to see how the
model works can explore the model more easily with pre-loaded data. For those wishing to use
their own data and are more familiar with the app, upload buttons are provided on the
input page for selection rates, mutation rates, and initial distribution.
How were the specific preset values on the input page chosen?
Mutation rates are based on experimental estimates for actual genes, assigned randomly to genes in our app.
The initial distribution vectors
were randomly Dirichlet sampled once for each gene and the selection rate vectors were randomly uniform
sampled once per gene, within acceptable selection rate limits based on the gene size. The number
of generations
n
is fixed at 220 regardless of the number of genes chosen.
Why is there an option to input selection per gene?
The vector size for selection without the option of selection per gene is length 2^(genes). This means that
fitness values can be chosen per phasotype. If the checkbox to input selection per gene is turned on,
a new (genes x 2) matrix appears in which you can amend selection per gene.
The app then applies a
tensor product
on this matrix to produce the values in the selection
vector of phasotypes. This can be seen as a multiplication of all phasotype combinations in the first
matrix (e.g. all 0 values multiplied
in the selection per gene matrix for three genes will equal the value seen in the 000 cell in the vector of
phasotypes).
The choice of using selection per gene may be the more desirable
option if you have an idea of selective values per gene, or which to save time with manual input of values.
However, the vector of phasotypes is then just a collection of products, and so better accuracy in results may be
attained if individual cells in this vector could be refined. Hence, the vector of phasotypes can still be edited
even if tensor calculations are made.
Regardless of whether the checkbox is checked on or off, the calculations performed when the
model is run is always from the phasotype vector.
Why does the app take a while to run?
The main reasons are either because the amount of genes chosen is quite large, or that the model is performing
a lot of calculations due to slow convergence to the stationary distribution.
To address the first issue in more detail, the computational cost increases at an exponential
rate when we look at more genes. This is evident in the vector sizes for the initial distribution and selection;
the vector sizes for each are of length 2^(genes), so for four genes they are length-16 vectors, but a subtle
change to six genes make them length-64 vectors. This is quite a large increase and is
more expensive for the model.
To address the second issue in more detail, we mentioned in the
How does the model work?
question
that the model in this app resembles a nonlinear Markov chain. One of the results of this assumption is that
the distribution at time
n
is dependent on the distribution at time
n-1
. Therefore, every generation up to the one chosen on the input page must be calculated, and also beyond that
to find the time to stationary distribution. Slow changes in distribution may be due to low mutation rates,
low (or no)
selection on phasotypes, or high numbers of genes.
The stationary distribution for this app is taken (by default) to be when the Euclidean distance between the
distribution at time
n
and the distribution at time
n-1
is lower than 10^(-5).
Why does my stationary distribution reach a limit?
The reason for large amounts of generations in the model is explained in the question above. To prevent excessive
waiting, the model stops automatically at a default value of 50,000 generations.
How can I change the settings of the app to make the run time shorter/longer?
In relation to the answers from the two previous questions, the following amendments can be made
(please run the model again after making amendments):
If the stationary distribution is reached in less generations than the number of generations
chosen on the input page, the final distribution returned will be the same as the stationary
distribution returned (as a time saving method). Therefore, if the tolerance here is chosen to be too low,
the final distribution (and by nature the stationary distribution) may not be accurate.
If the maximum generation number is less than the number of generations chosen in the input page,
the final distribution returned will only be up to the maximum generation number, unless stationarity
is achieved before then
What were your discoveries from the model?
We first tested the model using experimental
Campylobacter jejuni
data provided by Dr Bayliss' lab
at the University of Leicester. Estimates of the mutation rates are known from specially designed
experiments. From
in vitro
bacteria evolution experiments, we have sample
distributions at the start of the experiment (inoculum) and at the end of the experiment. The number of
generations
n
is estimated from the time length of the experiment, which was 220 for our data.
The fitness parameters are not directly observable. We developed an efficient method for checking
whether the mutation only model (i.e., when all the fitness parameters are equal) can explain the data.
We showed that the behaviour of some of the 28 phase variable genes of
Campylobacter jejuni
is
consistent with the mutation mechanism whilst there are other genes whose evolution cannot be explained
by mutations alone. For the latter, we need to estimate fitness parameters of our selection model.
The model was applied to the experimental data of such genes which may exhibit selection. With specific
choices for the selection vector, the final distribution of the model after 220 generations closely resembled
that of the actual data final distribution. This shows that our model provides one possible explanation
for the distributions we see in the data.
