## Lecture 10: Alignment - Introduction and Errors

Monday 2 May 2016

Scribed by Nattapoom Asavareongchai and revised by the course staff

## Topics

In the previous lecture, we learned about the multibridging algorithm, which was an improved k-mer based algorithm. We then looked at read-overlap graphs and the greedy and not-so-greedy algorithms associated with the graph. In this lecture we will consider the different types of errors that may be present and how they affect the algorithms we have derived. We will then move on to discuss the alignment problem.

## Assembly algorithms in practice

Recall that we have derived 2 classes of algorithms or 2 paradigms used for DNA assembly:

1. k-mer based algorithms:
• de Bruijin graph
• simple bridging & multibridging
• greedy algorithm
• not-so-greedy algorithm

Importantly, how are these algorithms used in practice?

The de Bruijin graph algorithm is usually used for short read assembly. Read-overlap graph algorithms are usually used for long reads (e.g. from Sanger sequencing).

The algorithms generally used for long read assembly are variants of the String graph. Standard assemblers that use these algorithms include the Celera assembler (and its succesor Canu) and the HGAP assembler (and its successor FALCON). Recently, assemblers like Miniasm have been developed based on ideas derived from the greedy algorithm to generate quick assemblies.

In practice assembling the whole DNA genome is impossible, so how well do the algorithms we have mentioned perform? What if we do not have enough reads/data?

• In the case of the de Bruijin graph algorithm, if there are repeats that cannot be resolved because of the lack of data, the algorithm does not break down and data on these repeats can be kept. Basically, the algorithm can take note that these particular repeats have not been resolved. If in the future more data is obtained and a read that is able to resolve the repeats is found, the algorithm can easily use stored data to improve past results. This holds true for the multibridging algorithm to a large extent as well. The most common scenario under which the algorithm can break down is the case where there is a triple repeat with at least one copy bridged by a read and at least one copy not bridged by reads (and some nested repeat setups).
• In contrast, the greedy and not-so-greedy algorithms do not keep information about unresolved repeats. This is because they are aggressive algorithms that rely on pruning the graph. Data are thrown out as much as possible to simplify the problem. The practical string graph based algorithms use some interesting heuristics to keep track of long repeats, reducing the amount of mis-assemblies.

## Errors

Our discussion in earlier lectures assume perfect reads with no errors; however, this is not the case in practice. There are always errors present during sequencing and base-calling. We want to therefore answer the question: What is the impact of errors on assembly? How do we account for these errors in the algorithms we have presented?

### Type of Errors

There are three types of errors:

1. Substitution
2. Insertion
3. Deletion

E.g. Suppose we have an underlying sequence: AGGCTTAAT. The following are examples of the 3 types of errors.

1. Substitution error: AGGCCTTAAT
2. Insertion error: AGGCTTAAAT
3. Deletion error: AGGCTT-AT

Second generation technologies like Illumina have mainly substitution errors at a rate of around 1-2%

Third and fourth generation technologies like PacBio and Nanopore have no PCR, and hence the error rate is $\approx$ 10-15% with mainly insertion and deletion errors.

### Impact of errors on greedy algorithm

Lets first consider the greedy algorithm and how errors will impact its performance. Recall that this algorithm merges the two reads with the longest overlap.

If there are errors present, we won’t be able to find a perfect overlap and the longest overlap may be corrupted by errors (and would be considered not an overlap by the algorithm).

To account for the errors, not only do we account for perfect overlap, but we must also consider approximate overlap. For example, an approximate overlap of 99% would mean that 99% of the overlapped nucleotides match while the other 1% do not. This percentage depends on the error rate. The approximate percentage should intuitively be = (1 - 2 x error rate). For example, a 1% error rate may cause 1% error on each side of the overlapped reads, thus there would be a possibility of a 2% error. The approximate percentage overlap would then be 98%.

### Impact of errors on k-mer algorithms

Now lets consider the class of k-mer algorithms. We will use an example to show how an error may affect our k-mer graph.

Consider a sequence of letters (we will use the English alphabet instead of A,G,C,T in this case) and the two reads of length 5:

Here, L = 5 and k = 3. The k-mers from the 2 reads are:

We can therefore create a k-mer graph from this 2 reads as follow:

Now suppose we have a third read, OWORL. But this read is corrupted and has an error, so we read OWXRL instead. The k-mers for this read are OWX, WXR, XRL. Therefore, combining this third read we have:

The error read creates a new path in the k-mer graph that branches out and may then be connected back to the main path. This branched out path is called a bubble. In practice, our dataset is large and we have a large number of reads. We may have both OWORL and OWXRL reads in our dataset. With bubbles, we will have two possible paths. This would create ambiguity in finding the Eulerian path from the k-mer graph.

There are methods used to address these:

1. We can align the two possible paths and choose the path that occurs more often. Since the error path is rare, there is a higher probability that the correct path will occur more often. Thus, choosing the more common path would eliminate the error bubble.
2. We can filter out reads that contain errors before actually creating our k-mer graph.

To find reads that contain errors in our dataset, we must consider the probability of correct reads occurring in our dataset as well as the probability of incorrect reads (errors) occurring.

So how many times will a particular k-mer appear, assuming that there are no error?

If we have k = 20, L = 100, and coverage = 30,

If we have an error rate of 1%,

Therefore,

Thus, a k-mer of length 20 will occur 20 times on average with no errors out of the 24 times that it may appear. Now consider the case with 1 error:

Therefore

Approximately 1 out of the 24 k-mer that occurs will have an error. We can therefore easily find the k-mer that rarely appears or in few numbers and discard them as error reads. We can set a threshold on the number of occurrences of a particular k-mer for it to be a valid k-mer.

Now lets consider a higher error rate of 15%. Doing the same calculation we have that:

A k-mer with no errors will appear rarely as well in this case. This means that it is very hard to tell if a k-mer that appears once is an error k-mer or a valid one. We won’t be able to set a threshold and pre-filtering breaks down at large error rates. We can prevent breakdowns by setting smaller k values (using smaller k-mers), but this approach may not be practical because of the huge size of our resulting k-mer dataset. Therefore with higher error rate, people usually do not use k-mer based algorithms and instead will choose to use read-overlap graph algorithms which do not have this problem.

## Alignment

In past lectures we have talked about assembling the genome from reads; however, before assembly we need to first merge two reads by aligning them with parts that overlap. We will now discuss how we solve this alignment problem. Because of errors, we will consider an algorithm that does an approximate alignment to find overlapping reads. There are two important problems where alignment is useful:

1. Finding overlapping reads for a de novo assembly.
2. Comparing reads with a reference genome to see which region the read belongs to. In turn we can find new variations and differences in the reads from the reference genome.

### Edit Distance & Dynamic Programming

To solve the alignment problem, we want to find the best way to align two strings by taking into account the three different types of errors we discussed earlier. Consider two strings $x$ and $y$:

In order to measure how much these two strings overlap, we will consider the edit distance of the two strings. Edit distance is defined as the the minimum # of insertion, deletion and substitutions necessary to get from string x to string y. Using the x and y given, we will denote a deletion by a -, and a substitution by a red letter.

x = GCGTATGTGGCTA-ACGC
y = GC-TATGCGGCTATACGC

We see from our example that we have two insertion/deletions at the 3rd and 14th letters of the sequence and a substitution at the 8th letter. Therefore there are 3 edits, resulting in an edit distance of 3.

One could compute edit distance between two strings by computing all combinations of insertions, deletions, and substitutions on the first string until we arrive at the second string. This approach will take exponential time in terms of the length of the string and is not efficient. An efficient solution would be to use Dynamic Programming.

An algorithm to find the edit distance would require us to break down the problem into subproblems. Using solutions from the subproblems, we can then build a working solution for the whole problem. We will discuss this algorithm next lecture.