**April 28, 2020**

In this class, our goal is to implement the Needleman-Wunsch algorithm in Python. You can read more about the Needleman-Wunsch algorithm on Wikipedia. The Wikipedia page contains pseudo-code which you might find helpful.

Since the Needleman-Wunsch algorithm works on a matrix of scores, it is helpful to discuss how to represent a matrix in Python. We can do so by creating a list of lists. Let's say we want to make a matrix that looks like this:

1 | 3 | 5 | 7 |

2 | 3 | 4 | 5 |

5 | 2 | 20 | 3 |

In Python, we would create it as follows:

In [1]:

```
my_matrix = []
# Fill out the 0th row
my_matrix.append([1, 3, 5, 7])
# Fill out the 1st row
my_matrix.append([2, 3, 4, 5])
# Fill out the 2nd row
my_matrix.append([5, 2, 20, 3])
print(my_matrix)
```

However, when we print the matrix, it is not easy to read, because it is shown as a list of lists. So it is helpful to define a function `print_matrix()`

that prints a matrix in a nicer form:

In [2]:

```
def print_matrix(mat):
# Loop over all rows
for i in range(0, len(mat)):
print("[", end = "")
# Loop over each column in row i
for j in range(0, len(mat[i])):
# Print out the value in row i, column j
print(mat[i][j], end = "")
# Only add a tab if we're not in the last column
if j != len(mat[i]) - 1:
print("\t", end = "")
print("]\n")
print_matrix(my_matrix)
```

To retrieve an indiviual value from a matrix, we use code of the form `my_matrix[row][column]`

. So, to retrieve for example the value in the 0th column of the 2nd row, we write:

In [3]:

```
print("The value in the 2nd row and the 0th column is:", my_matrix[2][0])
```

It will also be helpful to have a function that can create a matrix of zeros.

In [4]:

```
# A function for making a matrix of zeroes
def zeros(rows, cols):
return [[0 for i in range(cols)] for j in range(rows)]
zero_matrix = zeros(3, 5) # make zero matrix with 3 rows and 5 columns
print_matrix(zero_matrix)
```

**Problem 1:**

Write a function that takes two sequences as input and returns a matrix of Needleman-Wunsch scores. You **do not** have to do the back-tracing, just fill out the matrix.

Break the problem down into as many small steps as possible. Here are a few hints:

- Before you calculate any scores, make an empty matrix of the appropriate size using the
`zeros()`

function defined below. - Fill out the 0th row and 0th column before you calculate any other scores.
- The
`max()`

function will return the maximum value from a list of values. For example`max(1, 7, 3)`

will return`7`

. - Make liberal use of the
`range()`

function. - Use the
`print_matrix()`

function to print out your matrix as frequently as possible. Always make sure that your code is doing what you think it's doing! - Remember, in Python, we start counting from 0.

In [5]:

```
# Use these values to calculate scores
gap_penalty = -1
match_award = 1
mismatch_penalty = -1
# Make a score matrix with these two sequences
seq1 = "ATTACA"
seq2 = "ATGCT"
# A function for determining the score between any two bases in alignment
def match_score(alpha, beta):
if alpha == beta:
return match_award
elif alpha == '-' or beta == '-':
return gap_penalty
else:
return mismatch_penalty
# The function that actually fills out the matrix of scores
def needleman_wunsch(seq1, seq2):
# length of the two sequences
n = len(seq1)
m = len(seq2)
# Generate matrix of zeros to store scores
score = zeros(m+1, n+1)
# Calculate score table
# Fill out first column
for i in range(0, m + 1):
score[i][0] = gap_penalty * i
# Fill out first row
for j in range(0, n + 1):
score[0][j] = gap_penalty * j
# Fill out all other values in the score matrix
for i in range(1, m + 1):
for j in range(1, n + 1):
# Calculate the score by checking the top, left, and diagonal cells
match = score[i - 1][j - 1] + match_score(seq1[j-1], seq2[i-1])
delete = score[i - 1][j] + gap_penalty
insert = score[i][j - 1] + gap_penalty
# Record the maximum score from the three possible scores calculated above
score[i][j] = max(match, delete, insert)
return score
# Test out the needleman_wunsch() function
print_matrix(needleman_wunsch(seq1, seq2))
```

**Problem 2:**

Modify your code from Problem 1 to back-trace through the score matrix and print out the final alignment. **Hint:** For the back-tracing, you'll want to use a `while`

loop (or several of them).

In [6]:

```
def nw_backtrace(seq1, seq2):
# Calculate score table using the function we wrote earlier
score = needleman_wunsch(seq1, seq2)
# Create variables to store alignment
align1 = ""
align2 = ""
# Start from the bottom right cell in matrix
i = len(seq2)
j = len(seq1)
# We'll use i and j to keep track of where we are in the matrix, just like above
while i > 0 and j > 0: # end touching the top or the left edge
score_current = score[i][j]
score_diagonal = score[i-1][j-1]
score_up = score[i][j-1]
score_left = score[i-1][j]
# Check to figure out which cell the current score was calculated from,
# then update i and j to correspond to that cell.
if score_current == score_diagonal + match_score(seq1[j-1], seq2[i-1]):
align1 += seq1[j-1]
align2 += seq2[i-1]
i -= 1
j -= 1
elif score_current == score_up + gap_penalty:
align1 += seq1[j-1]
align2 += '-'
j -= 1
elif score_current == score_left + gap_penalty:
align1 += '-'
align2 += seq2[i-1]
i -= 1
# Finish tracing up to the top left cell
while j > 0:
align1 += seq1[j-1]
align2 += '-'
j -= 1
while i > 0:
align1 += '-'
align2 += seq2[i-1]
i -= 1
# Since we traversed the score matrix from the bottom right, our two sequences will be reversed.
# These two lines reverse the order of the characters in each sequence.
align1 = align1[::-1]
align2 = align2[::-1]
return(align1, align2)
output1, output2 = nw_backtrace(seq1, seq2)
print(output1 + "\n" + output2)
```