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@ -6,41 +6,114 @@
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**** Origin of replication (ori)
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Locating an ori is key for gene therapy (e.g. viral vectors), to introduce a theraupetic gene.
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Locating an ori is key for gene therapy (e.g. viral vectors), to introduce a theraupetic gene.
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**** Computational approaches to find ori in Vibrio Cholerae
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***** Exercise: find Pattern
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We'll look for the *DnaA box* sequence, using a sliding window, in that case we will use the function [[./Code/PatternCount.py][PatternCount]] to find out how many times
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does a sequence appear in the genome.
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We'll look for the *DnaA box* sequence, using a sliding window, in that case we will use this function to find out how many times
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does a sequence appear in the genome:
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For the second part, we're going to calculate the frequency map of the sequences of length /k/, for that purpose we'll use [[./Code/FrequentWords.py][FrequentWords]]
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#+BEGIN_SRC python
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def PatternCount(Text, Pattern):
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count = 0
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for i in range(len(Text)-len(Pattern)+1):
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if Text[i:i+len(Pattern)] == Pattern:
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count = count+1
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return count
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#+END_SRC
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For the second part, we're going to calculate the frequency map of the sequences
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of length /k/, for that purpose we'll use:
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#+BEGIN_SRC python
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def FrequentWords(Text, k):
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words = []
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freq = FrequencyMap(Text, k)
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m = max(freq.values())
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for key in freq:
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if freq[key] == m:
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words.append(key)
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return words
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def FrequencyMap(Text, k):
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freq = {}
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n = len(Text)
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for i in range(n - k + 1):
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Pattern = Text[i:i + k]
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freq[Pattern] = 0
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for i in range(n - k + 1):
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Pattern = Text[i:i + k]
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freq[Pattern] += 1
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return freq
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#+END_SRC
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***** Exercise: Find the reverse complement of a sequence
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We're going to generate the reverse complement of a sequence, which is the complement of a sequence, read in the same direction (5' -> 3').
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In this case, we're going to use [[./Code/ReverseComplement.py][ReverseComplement]]
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After using our function on the /Vibrio Cholerae's/ genome, we realize that some of the frequent /k-mers/ are reverse complements of other frequent ones.
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We're going to generate the reverse complement of a sequence, which is the complement of a sequence, read in the same direction (5' -> 3').
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In this case, we're going to use:
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#+BEGIN_SRC python
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def ReverseComplement(Pattern):
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Pattern = Reverse(Pattern)
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Pattern = Complement(Pattern)
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return Pattern
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def Reverse(Pattern):
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reversed = Pattern[::-1]
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return reversed
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def Complement(Pattern):
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compl = ""
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complement_letters = {"A": "T", "T": "A", "C": "G", "G": "C"}
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for char in Pattern:
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compl += complement_letters[char]
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return compl
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#+END_SRC
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After using our function on the /Vibrio Cholerae's/ genome, we realize that some of the frequent /k-mers/ are reverse complements of other frequent ones.
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***** Exercise: Find a subsequence within a sequence
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We're going to find the ocurrences of a subsquence inside a sequence, and save the index of the first letter in the sequence.
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This time, we'll use [[./Code/PatternMatching.py][PatternMatching]]
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After using our function on the /Vibrio Cholerae's/ genome, we find out that the /9-mers/ with the highest frequency appear in cluster.
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This is strong statistical evidence that our subsequences are /DnaA boxes/.
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We're going to find the ocurrences of a subsquence inside a sequence, and save the index of the first letter in the sequence.
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This time, we'll use:
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#+BEGIN_SRC python
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def PatternMatching(Pattern, Genome):
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positions = []
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for i in range(len(Genome)-len(Pattern)+1):
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if Genome[i:i+len(Pattern)] == Pattern:
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positions.append(i)
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return positions
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#+END_SRC
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After using our function on the /Vibrio Cholerae's/ genome, we find out that the /9-mers/ with the highest frequency appear in cluster.
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This is strong statistical evidence that our subsequences are /DnaA boxes/.
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**** Computational approaches to find ori in any bacteria
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Now that we're pretty confident about the /DnaA boxes/ sequences that we found, we are going to check if they are a common pattern in the rest of bacterias.
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We're going to find the ocurrences of the sequences in /Thermotoga petrophila/ using [[./Code/PatternCount.py][PatternCount]]
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Now that we're pretty confident about the /DnaA boxes/ sequences that we found, we are going to check if they are a common pattern in the rest of bacterias.
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We're going to find the ocurrences of the sequences in /Thermotoga petrophila/
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with:
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After the execution, we observe that there are *no* ocurrences of the sequences found in /Vibrio Cholerae/.
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We can conclude that different bacterias have different /DnaA boxes/.
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#+BEGIN_SRC python
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def PatternCount(Text, Pattern):
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count = 0
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for i in range(len(Text)-len(Pattern)+1):
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if Text[i:i+len(Pattern)] == Pattern:
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count = count+1
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return count
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#+END_SRC
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We have to try another computational approach then, find clusters of /k-mers/ repeated in a small interval.
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After the execution, we observe that there are *no* ocurrences of the sequences found in /Vibrio Cholerae/.
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We can conclude that different bacterias have different /DnaA boxes/.
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We have to try another computational approach then, find clusters of /k-mers/ repeated in a small interval.
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** Week 2
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@ -48,70 +121,193 @@
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**** Replication process
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The /DNA polymerases/ start replicating while the parent strands are unraveling.
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On the lagging strand, the DNA polymerase waits until the replication fork opens around 2000 nucleotides, and because of that it forms Okazaki fragments.
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We need 1 primer for the leading strand and 1 primer per Okazaki fragment for the lagging strand.
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While the Okazaki fragments are being synthetized, a /DNA ligase/ starts joining the fragments together.
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The /DNA polymerases/ start replicating while the parent strands are unraveling.
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On the lagging strand, the DNA polymerase waits until the replication fork opens around 2000 nucleotides, and because of that it forms Okazaki fragments.
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We need 1 primer for the leading strand and 1 primer per Okazaki fragment for the lagging strand.
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While the Okazaki fragments are being synthetized, a /DNA ligase/ starts joining the fragments together.
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**** Computational approach to find ori using deamination
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As the lagging strand is always waiting for the helicase to go forward, the lagging strand is mostly in single-stranded configuration, which is more prone to mutations.
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One frequent form of mutation is *deamination*, a process that causes cytosine to convert into thymine. This means that cytosine is more frequent in half of the genome.
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As the lagging strand is always waiting for the helicase to go forward, the lagging strand is mostly in single-stranded configuration, which is more prone to mutations.
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One frequent form of mutation is *deamination*, a process that causes cytosine to convert into thymine. This means that cytosine is more frequent in half of the genome.
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***** Exercise: count the ocurrences of cytosine
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We're going to count the ocurrences of the bases in a genome and include them in a symbol array, for that purpose we'll use [[./Code/SymbolArray.py][SymbolArray]]
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After executing the program, we realize that the algorithm is too inefficient.
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We're going to count the ocurrences of the bases in a genome and include them in
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a symbol array, for that purpose we'll use:
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#+BEGIN_SRC python
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def SymbolArray(Genome, symbol):
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array = {}
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n = len(Genome)
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ExtendedGenome = Genome + Genome[0:n//2]
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for i in range(n):
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array[i] = PatternCount(ExtendedGenome[i:i+(n//2)], symbol)
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return array
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def PatternCount(Text, Pattern):
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count = 0
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for i in range(len(Text)-len(Pattern)+1):
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if Text[i:i+len(Pattern)] == Pattern:
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count = count+1
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return count
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#+END_SRC
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After executing the program, we realize that the algorithm is too inefficient.
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***** Exercise: find a better algorithm for the previous exercise
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This time, we are going to evaluate an element /i+1/, using the element /i/. We'll use [[./Code/FasterSymbolArray.py][FasterSymbolArray]] to achieve this
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After executing the program we see that it's a viable algorithm, with a complexity of /O(n)/ instead of the previous /O(n²)/.
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In /Escherichia Coli/ we plotted the result of our program:
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This time, we are going to evaluate an element /i+1/, using the element /i/.
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We'll use the following algorithm:
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#+CAPTION: Symbol array for Cytosine in E. Coli Genome]
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[[./Assets/e-coli.png]]
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#+BEGIN_SRC python
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def FasterSymbolArray(Genome, symbol):
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array = {}
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n = len(Genome)
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ExtendedGenome = Genome + Genome[0:n//2]
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array[0] = PatternCount(symbol, Genome[0:n//2])
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for i in range(1, n):
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array[i] = array[i-1]
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if ExtendedGenome[i-1] == symbol:
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array[i] = array[i]-1
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if ExtendedGenome[i+(n//2)-1] == symbol:
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array[i] = array[i]+1
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return array
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From that graph, we conclude that ori is located around position 4000000, because that's where the Cytosine concentration is the lowest,
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which indicates that the region stays single-stranded for the longest time.
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def PatternCount(Text, Pattern):
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count = 0
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for i in range(len(Text)-len(Pattern)+1):
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if Text[i:i+len(Pattern)] == Pattern:
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count = count+1
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return count
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#+END_SRC
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After executing the program we see that it's a viable algorithm, with a complexity of /O(n)/ instead of the previous /O(n²)/.
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In /Escherichia Coli/ we plotted the result of our program:
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#+CAPTION: Symbol array for Cytosine in E. Coli Genome]
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[[./Assets/e-coli.png]]
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From that graph, we conclude that ori is located around position 4000000, because that's where the Cytosine concentration is the lowest,
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which indicates that the region stays single-stranded for the longest time.
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**** The Skew Diagram
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Usually scientists measure the difference between /G - C/, which is *higher on the lagging strand* and *lower on the leading strand*.
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Usually scientists measure the difference between /G - C/, which is *higher on the lagging strand* and *lower on the leading strand*.
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***** Exercise: Synthetize a Skew Array
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We're going to make a Skew Diagram, for that we'll first need a Skew Array. For that purpose we wrote [[./Code/SkewArray.py][SkewArray]]
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We can see the utility of a Skew Diagram looking at the one from /Escherichia Coli/:
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We're going to make a Skew Diagram, for that we'll first need a Skew Array. For
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that purpose we wrote:
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#+CAPTION: Symbol array for Cytosine in E. Coli Genome]
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[[./Assets/skew_diagram.png]]
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#+BEGIN_SRC python
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def SkewArray(Genome):
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Skew = []
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Skew.append(0)
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for i in range(0, len(Genome)):
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if Genome[i] == "G":
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Skew.append(Skew[i] + 1)
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elif Genome[i] == "C":
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Skew.append(Skew[i] - 1)
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else:
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Skew.append(Skew[i])
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return Skew
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#+END_SRC
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Ori should be located where the skew is at its minimum value.
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We can see the utility of a Skew Diagram looking at the one from /Escherichia Coli/:
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#+CAPTION: Symbol array for Cytosine in E. Coli Genome]
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[[./Assets/skew_diagram.png]]
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Ori should be located where the skew is at its minimum value.
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***** Exercise: Efficient algorithm for locating ori
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Now that we know more about ori's skew value, we're going to construct a better algorithm to find it. We'll do that in [[./Code/MinimumSkew.py][MinimumSkew]]
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Now that we know more about ori's skew value, we're going to construct a better
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algorithm to find it:
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#+BEGIN_SRC python
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def MinimumSkew(Genome):
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positions = []
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skew = SkewArray(Genome)
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minimum = min(skew)
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return [i for i in range(0, len(Genome)) if skew[i] == minimum]
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def SkewArray(Genome):
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Skew = []
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Skew.append(0)
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for i in range(0, len(Genome)):
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if Genome[i] == "G":
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Skew.append(Skew[i] + 1)
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elif Genome[i] == "C":
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Skew.append(Skew[i] - 1)
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else:
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Skew.append(Skew[i])
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return Skew
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#+END_SRC
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**** Finding /DnaA boxes/
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When we look for /DnaA boxes/ in the minimal skew region, we can't find highly repeated /9-mers/ in /Escherichia Coli/.
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But we find approximate sequences that are similar to our /9-mers/ and only differ in 1 nucleotide.
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When we look for /DnaA boxes/ in the minimal skew region, we can't find highly repeated /9-mers/ in /Escherichia Coli/.
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But we find approximate sequences that are similar to our /9-mers/ and only differ in 1 nucleotide.
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***** Exercise: Calculate Hamming distance
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The Hamming distance is the number of mismatches between 2 strings, we'll solve this problem in [[./Code/HammingDistance][HammingDistance]]
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The Hamming distance is the number of mismatches between 2 strings, we'll solve this problem in [[./Code/HammingDistance][HammingDistance]]
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***** Exercise: Find approximate patterns
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Now that we have our Hamming distance, we have to find the approximate sequences. We'll do this in [[./Code/ApproximatePatternMatching.py][ApproximatePatternMatching.py]]
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Now that we have our Hamming distance, we have to find the approximate
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sequences:
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#+BEGIN_SRC python
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def ApproximatePatternMatching(Text, Pattern, d):
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positions = []
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for i in range(len(Text)-len(Pattern)+1):
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if Text[i:i+len(Pattern)] == Pattern:
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positions.append(i)
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elif HammingDistance(Text[i:i+len(Pattern)], Pattern) <= d:
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positions.append(i)
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return positions
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def HammingDistance(p, q):
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count = 0
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for i in range(0, len(p)):
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if p[i] != q[i]:
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count += 1
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return count
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#+END_SRC
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***** Exercise: Count the approximate patterns
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The final part is counting the approximate sequences, for that we'll use [[./Code/ApproximatePatternCount.py][ApproximatePatternCount.py]]
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The final part is counting the approximate sequences:
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After trying out our ApproximatePatternCount in the hypothesized ori region, we find a frequent /k-mer/ with its reverse complement in /Escherichia Coli/.
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We've finally found a computational method to find ori that seems correct.
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#+BEGIN_SRC python
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def ApproximatePatternCount(Pattern, Text, d):
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count = 0
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for i in range(len(Text)-len(Pattern)+1):
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if Text[i:i+len(Pattern)] == Pattern or HammingDistance(Text[i:i+len(Pattern)], Pattern) <= d:
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count += 1
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return count
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def HammingDistance(p, q):
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count = 0
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for i in range(0, len(p)):
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if p[i] != q[i]:
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count += 1
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return count
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#+END_SRC
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After trying out our ApproximatePatternCount in the hypothesized ori region, we find a frequent /k-mer/ with its reverse complement in /Escherichia Coli/.
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We've finally found a computational method to find ori that seems correct.
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** Week 3
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@ -123,7 +319,6 @@ Variation in gene expression permits the cell to keep track of time.
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***** Exercise: Find the most common nucleotides in each position
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We are going to create a *t x k* Motif Matrix, where *t* is the /k-mer/ string. In each position, we'll insert the most frequent nucleotide, in upper case,
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and the nucleotide in lower case (if there's no popular one).
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Our goal is to select the *most* conserved Matrix, i.e. the Matrix with the most upper case letters.
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@ -493,5 +688,5 @@ characteristic of Greedy Algorithms, they trade optimality for speed.
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** Vocabulary
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- k-mer: subsquences of length /k/ in a biological sequence
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- Frequency map: sequence --> frequency of the sequence
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- k-mer: subsquences of length /k/ in a biological sequence
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- Frequency map: sequence --> frequency of the sequence
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