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 Incoming links: Smith-Waterman algorithm

Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / C4 / g / Solution 2026-10-20

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Use local sequence alignment, such as the Smith-Waterman algorithm, to compare the globin fragment with the matching region of each full-length target. A global sequence alignment would penalize long target flanks absent from the fragment. For a linear gap penalty the dynamic programming recurrence includes a restart option:

H_{ij} = max {0, H_{i - 1, j - 1} + s (a_{i}, b_{j}), H_{i - 1, j} - g, H_{i, j - 1} - g} .

(1)

Initialize boundary entries to zero, start traceback at the largest entry anywhere, and stop at zero. If the entire query is known to be valid and must be retained, fitting sequence alignment is a suitable refinement: align the whole fragment while leaving target flanks unpenalized.

First check the sequence alphabet: the prompt calls the targets “genes”. If the FASTA format file contains DNA rather than protein, translate the relevant coding sequences into amino acids before a protein-to-protein sequence alignment. If coding frames are unknown, search translated frames. A file in FASTA format specifies storage syntax, not whether the contents are nucleotides or residues.

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Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 2 / C7 / a / Solution 2026-10-20

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Use the supplied BLOSUM62 entry for every diagonal move in the Smith-Waterman algorithm, while retaining the stipulated linear gap penalty of

- 1

. If

q_{i}

and

r_{j}

are the current amino acids, the dynamic programming recurrence is

H_{ij} = max {0, H_{i, j - 1} - 1, H_{i - 1, j} - 1, H_{i - 1, j - 1} + B (q_{i}, r_{j})} .

(1)

The zero option permits a new local start. The following original implementation also records the traceback coordinates needed in part (c); its tie ordering is left, up, diagonal, and an equal overall maximum replaces the previous best cell, as in the supplied convention:

def local_alignment(query,reference,alphabet,scores,gap=-1):
    lookup={aa:i for i,aa in enumerate(alphabet)}
    table=np.asarray(scores,dtype=int)
    height,width=len(query)+1,len(reference)+1
    H=np.zeros((height,width),dtype=int)
    back=np.zeros((height,width),dtype=np.int8)
    best=(0,0,0)
    for i,a in enumerate(query,1):
        for j,b in enumerate(reference,1):
            # Order implements the printed LEFT, UP, DIAGONAL tie priority.
            candidates=(0,H[i,j-1]+gap,H[i-1,j]+gap,
                        H[i-1,j-1]+int(table[lookup[a],lookup[b]]))
            value=max(candidates);H[i,j]=value
            back[i,j]=0 if value==0 else candidates.index(value)
            if value>=best[0]:best=(value,i,j)
    score,i,j=best;query_end,ref_end=i,j
    aq=[];ar=[]
    while H[i,j]>0:
        step=back[i,j]
        if step==1:aq.append('-');ar.append(reference[j-1]);j-=1
        elif step==2:aq.append(query[i-1]);ar.append('-');i-=1
        elif step==3:aq.append(query[i-1]);ar.append(reference[j-1]);i-=1;j-=1
    qa=''.join(reversed(aq));ra=''.join(reversed(ar));length=len(qa)
    identities=sum(a==b and a!='-' for a,b in zip(qa,ra))
    similarities=sum(a!='-' and b!='-' and table[lookup[a],lookup[b]]>0 for a,b in zip(qa,ra))
    gaps=sum(a=='-' or b=='-' for a,b in zip(qa,ra))
    return dict(query=qa,reference=ra,score=int(score),query_start=i+1,query_end=query_end,
                reference_start=j+1,reference_end=ref_end,query_length=len(query),reference_length=len(reference),
                alignment_length=length,identities=int(identities),similarities=int(similarities),gaps=int(gaps),
                identity_percent=100*identities/length if length else 0.,
                similarity_percent=100*similarities/length if length else 0.,gap_percent=100*gaps/length if length else 0.)

queries = {"baboon": baboon, "squirrel_monkey": squirrel_monkey,
           "dingo": dingo, "goldfish": goldfish}
alignments = {
    name: local_alignment(seq, reference_epo_receptor_human, amino_acids, bmat)
    for name, seq in queries.items()
}
for name, result in alignments.items():
    print(name, "score", result["score"])
    print(result["query"])
    print(result["reference"])

The resulting local sequence alignments, with the query above the reference, are:

baboon; score 314

SLLTPSDLDPLILTLSLILVLIVVLLTVLALLSHRRALKQKIWPGIPSPESEFEGLFTTHKG-FQL
SLLTPSDLDPLILTLSLILVVILVLLTVLALLSHRRALKQKIWPGIPSPESEFEGLFTTHKGNFQL

squirrel monkey; score 278

VCFWEEAASAGMGPDNYSFSYQL-DNEPWKR-CRLHQTPTARGVVRFWCSLPRADTSS
VCFWEEAASAGVGPGNYSFSYQLED-EPWK-LCRLHQAPTARGAVRFWCSLPTADTSS

dingo; score 301

MNHLW--THLWPG-VGSLCLLLAGAAWASLPKP-L-DPKFESKAALLAARAPEELLCFTERLEDLVCFWEE
MDHL-GAS-LWP-QVGSLCLLLAGAAWAP-P-PNLPDPKFESKAALLAARGPEELLCFTERLEDLVCFWEE

goldfish; score 190

L--LVLSIGLILC--L-LSLTVI-L-SHQKFLLK-KLWPDIPTPEHKFPGLFTVYKGDFKE-WMSH-NNGSM
LDPLILTLSLILVVILVL-LTVLALLSHRR-ALKQKIWPGIPSPESEFEGLFTTHKGNF-QLWL-YQNDGCL

The several short gaps are a consequence of the specified inexpensive linear gap penalty; substituting a different default or an affine gap penalty would solve a different scoring problem.

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