Preface: CpG islands

It is known that due to biochemical considerations that CpG, the pair of nocleotides C and G, appearing successively, in this order, along one DNA starnd, is relatively rare in DNA sequences, excluding particular sub-sequences, which are several hundreds of nucleotides long, where the couple CpG is more frequent. These sub-sequences, called CpG islands, are known to appear in the biologically more significant parts of the genome. The ability to identify CpG islands in the DNA will therefore help us spot the more significant regions of interest along the genome.

Problem 6.1   Identifying a CpG island.
INPUT: A short DNA sequence $X=(x_{1},\ldots,x_{L}) \in \Sigma^{*}$ (where $\Sigma = \{\text{A,C,G,T}\}$).
QUESTION: Decide whether X is a CpG island.

We can approach such problems using a Markov chain model. Let us denote for each $s,t \in \Sigma$ the transition probability:

$$a_{st} \equiv P(x_{i}=t \vert x_{i-1}=s)$$ (6.1)

We assume that $\{x_{i}\}$ is a random process with a memory of length 1, i.e., the value of the random variable x_i depends only on its predecessor x_i-1. Formally we can write:

$$\begin{split} \forall {s_{1},\ldots,s_{i} \in \Sigma} \quad... ..._{i} \vert x_{i-1}=s_{i-1}) = a_{s_{i-1},s_{i}} \end{split}$$ (6.2)

The probability of the whole sequence X will therefore be:

$$P(X) = p(x_{1}) \cdot \prod_{i=2}^{L}a_{x_{i-1},x_{i}}$$ (6.3)

We can also add fictitious $begin \, (=x_{0})$ and $end \, (=x_{L+1})$ symbols to simplify the formula, where $\forall_{s \in \Sigma} \, a_{0,s} \equiv p(s)$ is the background probability of the symbol s. Hence:

$$P(X) = \prod_{i=1}^{L}a_{x_{i-1},x_{i}}$$ (6.4)

Let a⁺_st denote the transition probability of $s,t \in \Sigma$ inside a CpG island and let a^-_st denote the transition probability outside a CpG island (see table 6.1 for the values of these probabilities, taken from [4] ). We can give a logarithmic likelihood score for the sequence X:

$$Score(X) = \log \frac{P(X \vert \text{CpG island})}{P(X \vert... ...1}^{L}\log\frac{a^{+}_{x_{i-1},x_{i}}}{a^{-}_{x_{i-1},x_{i}}}$$ (6.5)

The higher this score, the more likely is that X is a CpG Island.

 

Table 6.1: Transition probabilities inside/outside a CpG island

+	A	C	G	T	-	A	C	G	T
A	0.180	0.274	0.426	0.120	A	0.300	0.205	0.285	0.210
C	0.171	0.368	0.274	0.188	C	0.322	0.298	0.078	0.302
G	0.161	0.339	0.375	0.125	G	0.248	0.246	0.298	0.208
T	0.079	0.355	0.384	0.182	T	0.177	0.239	0.292	0.292

 

Problem 6.2   Locating CpG islands in a DNA sequence.
INPUT: A long DNA sequence $X=(x_{1},\ldots,x_{L}) \in \Sigma^{*}$.
QUESTION: Locate the CpG islands within X.

A naive approach for solving this problem will be to extract a sliding window $X^{k}=(x_{k+1},\ldots,x_{k+\ell})$ of a given length $\ell$ (where $\ell \ll L$, usually several hundreds long, and $1 \le k \le L-\ell$) to the sequence and calculate Score(X^k) for each one of the resulting sub-sequences. Sub-sequences that receive positive scores are potential CpG islands.
The main disadvantage in this algorithm is that we have no information about the lengths of the islands, while the algorithm suggested above assumes that those islands are at least $\ell$ nucleotides long. Should we use a value of $\ell$ which is too large, the CpG islands would be short sub-strings of our windows, and the score we give those windows may not be high enough. A better approach to such problems is described in the following section.