**Source:** Uçarlı, Cüneyt, Liam J. McGuffin, Süleyman Çaputlu, Andres Aravena, and Filiz Gürel. *“Genetic Diversity at the Dhn3 Locus in Turkish Hordeum Spontaneum Populations with Comparative Structural Analyses.”* Scientific Reports (2016) https://doi.org/10.1038/srep20966.

- To study closely related genes or proteins
- to find conserved domains

- To find the evolutionary relationships
- the base of phylogenetic trees

- To identify shared patterns among related genes
- conserved sequences can be binding sites

We discussed *pairwise alignment*

- Global, using
*Needleman & Wunsch*algorithm- also for semi-global alignment

- Local, using
*Smith & Waterman*algorithm

We build a matrix with \(m_1\) rows and \(m_2\) columns

We write the sequence \(s_1\) in the rows,

and the sequence \(s_2\) in the columns

The computational cost is \(O(m_1 m_2)\)

(\(m_1\) is the length of \(s_1\), \(m_2\) is the length of \(s_2\))

To find the optimal alignment, we look for diagonals in the matrix that maximize the total score

Every cell \(M_{ij}\) in the matrix has initially the value \[M_{ij}=\text{Score}_2(s_{1}[i],s_{2}[j])\] where \(s_{1}[i]\) is the letter in position \(i\) of sequence \(s_1\),

and \(s_{2}[j]\) is the letter in position \(j\) of \(s_2\)

To aligning two sequences, we build a dot-plot matrix.

That is, a rectangle.

To align three sequences, we need a three-dimensional array.

That is, a cube.

Each cell \(M_{ijk}\) has value \[M_{ijk}=\text{Score}_3(s_{1}[i],s_{2}[j], s_{3}[k])\]

Usually, external gaps do not count, but internal gaps count

That is, these are semi-global alignments

Any path from a border to another border will be an alignment

We look for the *optimal* alignment

If the three sequences have length \(m_1, m_2,\) and \(m_3,\) then building the cube has cost \[O(m_1\cdot m_2\cdot m_3)\]

To simplify, we assume that all sequences have length \(m\)

Then the cost of three-wise alignment is \[O(m^3)\]

Following the same idea…

To align \(N\) sequences, we need a dot-plot in \(N\) dimensions

\[M_{i_1,\ldots,i_N}=\text{Score}_N(s_{1}[i_1],s_{2}[i_2],…,s_{N}[i_N])\]

Therefore, if the average sequence length is \(m,\) then the cost is \[O(m^N)\]

To fix ideas, assume that \(m=1000\)

(That is a typical size for a bacterial gene)

The computational cost is \(O(1000^N)\)

In other words, the cost is \(O(10^{3N})\)

Now assume that the computer can do one million comparisons each second

The number of seconds is then \[O(10^{3N-6})\]

Exercise: How many seconds will it take for 2, 4, 8, and 12 sequences?

Under these hypothesis we have this table

\(N\) | Seconds | In words |
---|---|---|

2 | \(10^0\) | 1 sec |

4 | \(10^6\) | 1 million seconds |

8 | \(10^{18}\) | 1 trillion/quintillion seconds |

12 | \(10^{30}\) | a lot of time |

Translate these numbers to days, years, etc.

(Approximate answer are OK. We only need one significant figure)

How do these numbers change if \(m\) changes?

What happens if the computers are 1000 times faster?

What is the largest multiple alignment that you can do in your life?

What is the largest number of sequences that can be aligned?

What can we do to align more sequences?

This is clearly too expensive, so we need heuristics

(i.e. solving a similar but simpler problem)

One common idea is to do a *progressive alignment*

- start with a pairwise alignment,

- and then align the rest one by one

There are several ways to simplify the original problem

Thus, there are many *approximate* solutions

The main differences are:

- How to decide what to align first?
- Can new sequences change the previous alignment?
- Can \(s_k\) change the alignment of \((s_1,…,s_{k-1})\)?

- How to use additional information (like 3D structure)?
- What is the formula for \(\text{Score}_k(s_{1}[i_1],…,s_{k}[i_k])\)?

- Clustal
- ClustalW, ClustalX, Clustal Omega

- T-Coffee
- 3D-Coffee

- MUSCLE
- MAFFT