<p>$C_k$ is defined as the number of reads containing a k-mer. The fraction of a read available to contain a k-mer is $(R-K+1)/R$, which is the number of k-mers in the read divided by its length. That times the nucleotide coverage ($C$) is then the expected coverage of the k-mer.</p>

CC BY-SA 3.0
2017-07-26T20:45:29.493Z
2017-07-26T20:45:29.493Z
Devon Ryan
<p>I was still puzzled from the answers, so I tried to calculate with all the steps. I take this definition "$C_k$ is the number of reads containing a k-mer." and corresponding definition for coverage ($C$): "$C$ is the number of reads covering a base".</p>

<p>Coverage is $C = \frac{T \cdot R}{L}$, where $T$ is total number of reads, $R$ is read length and $L$ is length of genome. Given the $C_k$ definition, $C_k = \frac{T (R - K + 1)}{L-K+1}$, where $R - K + 1$ is just number of kmers in a read, and $L-K+1$ is number of kmers in a genome. Then,</p>

<p>$$C_k = \frac{T (R - K + 1)}{L-K+1} = \frac{T (R - K + 1)}{L-K+1} \cdot \frac{R}{R} = \frac{R - K + 1}{R} \cdot \frac{T \cdot R}{L - K + 1}$$</p>

<p>since $L &gt;&gt; K$, we can approximate $L - K + 1 \approx L$, then we reduce the expression to</p>

<p>$$\frac{R - K + 1}{R} \cdot \frac{T \cdot R}{L} = \frac{R - K + 1}{R} \cdot C$$</p>

<p>which is the formula for $C_k$.</p>

CC BY-SA 3.0
2017-07-27T10:02:08.870Z
2017-07-27T10:54:57.303Z
2017-07-27T10:54:57.303Z
Kamil S Jaron
<p>Let $C$ be base coverage, $R$ is the length of reads and $K$ is the length of $k$-mer. Then $k$-mer coverage $C_k$ can be computed as $C_k = C\cdot(R - K + 1)/R$.</p>

<p>Could someone please explain why is this equation valid (I'm mostly confused as why it is divided by $R$)?</p>

<p>Source: <a href="http://www.ebi.ac.uk/~zerbino/velvet/Manual.pdf" rel="noreferrer">Velvet manual</a> Section 5.1</p>

<k-mer>
Formula for k-mer coverage
# Formula for k-mer coverage

(Asked by: username_0 on 2017-07-26)

Let $C$ be base coverage, $R$ is the length of reads and $K$ is the length of $k$-mer. Then $k$-mer coverage $C\_k$ can be computed as $C\_k = C\cdot(R - K + 1)/R$.

Could someone please explain why is this equation valid (I'm mostly confused as why it is divided by $R$)?

Source: [Velvet manual](http://www.ebi.ac.uk/~zerbino/velvet/Manual.pdf) Section 5.1

**Tags:** k-mer

---

## 2 answers

### Answer by username_1 on 2017-07-26. Score: 3 | Accepted answer

$C\_k$ is defined as the number of reads containing a k-mer. The fraction of a read available to contain a k-mer is $(R-K+1)/R$, which is the number of k-mers in the read divided by its length. That times the nucleotide coverage ($C$) is then the expected coverage of the k-mer.


---

### Answer by username_2 on 2017-07-27. Score: 5

I was still puzzled from the answers, so I tried to calculate with all the steps. I take this definition "$C\_k$ is the number of reads containing a k-mer." and corresponding definition for coverage ($C$): "$C$ is the number of reads covering a base".

Coverage is $C = \frac{T \cdot R}{L}$, where $T$ is total number of reads, $R$ is read length and $L$ is length of genome. Given the $C\_k$ definition, $C\_k = \frac{T (R - K + 1)}{L-K+1}$, where $R - K + 1$ is just number of kmers in a read, and $L-K+1$ is number of kmers in a genome. Then,

$$C\_k = \frac{T (R - K + 1)}{L-K+1} = \frac{T (R - K + 1)}{L-K+1} \cdot \frac{R}{R} = \frac{R - K + 1}{R} \cdot \frac{T \cdot R}{L - K + 1}$$

since $L >> K$, we can approximate $L - K + 1 \approx L$, then we reduce the expression to

$$\frac{R - K + 1}{R} \cdot \frac{T \cdot R}{L} = \frac{R - K + 1}{R} \cdot C$$

which is the formula for $C\_k$.


---

