Probabilistic Models for Mapping

Recall the following defintion:  A Poisson process of rate $\lambda$ is described by:

• A non decreasing function $N: R_{0}^{+} \rightarrow N$ where N(t) = number of events until time t
• N(0) = 0
• The number of events in disjoint intervals are independent
  

  $$P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac {({\lambda t})^{n}} { n! } \mbox{, for } n = 0,1,\ldots \mbox { and } s \geq 0$$ (2)

  
  
  As as consequence, distribution of the number of events in an interval is stationary, i.e. depends only on the length of the interval. The expected number of events in an interval of length t is given by $E(N(t)) = \lambda t$.

Denote:

• T_n = time between event n-1 and event n
• S₀ = 0
• $S_{\imath} = \sum_{i=1}^{i} T_{\imath}$

Recall that inter-event times in a Poisson process are i.i.d. random variables, exponentially distributed with parameter $\lambda$, i.e.,

$$Pr(T_{i} > t) = e^{- \lambda t}$$ (3)

If it is known that $n \geq 1$ events occurred in a Poisson process until time t, then the inter-arrival times $\{S_{1},S_{2},...,S_{n} \}$ are distributed uniformly and independently in [0,t]. Assume clone length L, genome length G, and choose N clones at random. What is the expected fraction of the genome covered by clones? For a random point b, and an arbitrary clone C the probability of the point b being included in the clone c is given by:

$$Pr(b \in c) = \frac {L} {G}$$ (4)

and therefore, the probability of b being out of all the clones is given by:

$$Pr(\forall c : b \notin c) = (1 - \frac {L} {G})^{N} = (1 - \frac {L} {G})^{G \frac {N} {G}} \sim e^{-\frac {NL} {G}}$$ (5)

with the last approximation being valid when $L \ll G$ and $N \ll G$.   The fraction

$$R=\frac{NL}{G}$$

is said to be the redundancy of the clone set.   The expected fraction of non-covered genome is given by

$$E(\mbox{fraction not covered}) = e^{-R}$$ (6)

where R is the redundancy of the clone set Table 9.2 shows that using redundancy factor of 2 to 5 gives a good coverage of the genome segment considered.

 

Table 9.1: Coverage of genome segment depending on redundancy factor

 

R	Coverage
1	0.63
2	0.865
3	0.95
4	0.98
5	0.993

Assume clone length = 1, and denote N = number of clones, R = redundancy factor, and assume that the clone starting positions follow a Poisson process with rate $\lambda$. We define a minimal overlap factor $\theta$ between clones to identify overlap, that is, two clones defined to overlap only if they share at least a $\theta$-length section. A set of clones covering a continuous segment of the genome, together with their physical distances is called a contig. Contigs are sometimes referred to as islands.