A8LastSlides_MT22
1 A genetic recombination model
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In most of our cells, we have two copies of each chromosome, one inherited
from our mother and one from our father. Gametes sperm and ova - contain
only one copy of each chromosome.
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During meiosis, the chromosomes are broken at certain random
"recombination" points, to form new chromosomes out of pieces of the
maternal and paternal chromosomes.
Genes occur at particular positions along the chromosome. In early genetic
research, biologists investigated the position of genes on chromosomes by studying
how often they were inherited together.
Here, genes \(b, c\) and \(d\) stay together but \(a\) is separated from them.
new chromosomes
As a simple model, represent the chromosome as a continuous line, and model the
recombination points as a Poisson process along it, of rate \(\lambda \). Let two points \(a\) and \(b\) on
the interval representing the location of two genes. Let \(x\) be the distance between
them.
The probability of seeing no crossover between \(a\) and \(b\) is
$$\mathbb{P}(N(a, b]=0)=e^{-\lambda x}$$
The probability to see an even number of crossovers between \(a\) and \(b\) :
$$
\begin{aligned}
\sum_{k=0}^{\infty} e^{-\lambda x} \frac{(\lambda x)^{2 k}}{(2 k) !} &=e^{-\lambda x}\left(1+\frac{(\lambda x)^{2}}{2 !}+\frac{(\lambda x)^{4}}{4 !}+\ldots\right) \\
&=e^{-\lambda x}\left(\frac{e^{\lambda x}+e^{-\lambda x}}{2}\right) \\
&=\frac{1+e^{-2 \lambda x}}{2} .
\end{aligned}
$$
If we observe that \(a\) and \(b\) are inherited together with probability \(p>1 / 2\), we can invert
this to estimate \(x\) by
$$x=-\frac{1}{2 \lambda} \log (2 p-1)$$
2 Poisson processes in higher dimensions
Consider a "process" of points in \(\mathbb {R}^{d}\). For a set \(S \subset \mathbb {R}^{d}\), let \(N(S)\) be the number of points falling in \(S\).
The process is a Poisson process of rate \(\color{red}\lambda \) if:
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If \(S_{1}, S_{2}, \ldots , S_{k}\) are disjoint sets, then the counts \(N\left (S_{1}\right ), N\left (S_{2}\right ), \ldots , N\left (S_{k}\right )\) are independent.
For any set \(S\) of area \(a, N(S) \sim \) Poisson \((\lambda a)\).
Beginning in June, 1944, London was the target of attacks by V-1 "flying bombs"
launched primarily from France. How precisely were these missiles targeted? Did the
impact sites in central London cluster more than would have been expected by
chance?
R. D. Clarke analysed the sites within a \(12 \mathrm {~km}\) by \(12 \mathrm {~km}\) heavily bombed region of central
London. He divided the region up into 576 squares of side length \(0.5 \mathrm {~km}\). The total number
of impacts was 537 .
Clarke compared the set of counts from each square with a Poisson distribution of
mean \(537 / 576\).
Distribution of bomb strikes on central London, 1944
Distribution of bomb strikes in an area of central London, 1944
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| | No. of flying bombs per square | Expected no. of squares (Poisson) | Actual no. of squares |
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| | 0 | 226.74 | 229 | | 1 | 211.39 | 211 |
| 2 | 98.54 | 93 | | 3 | 30.62 | 35 |
| 4 | 7.14 | 7 | | 5 and over | 1.57 | 1 |
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| | | 576.00 | 576 |
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3 Review of the course
4 What comes next?!
Part A:
Part B:
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Applied Probability
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Probability, Measure and Martingales
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Continuous Martingales and Stochastic Calculus
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Mathematical Models of Financial Derivatives
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Information Theory
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Stochastic Modelling of Biological Processes
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various Statistics courses
Part C: all sorts of options in probability, statistics, genetics, stochastic analysis,
combinatorics, machine learning...
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