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Parikh vectors
3USC_1 8PND_1 1EOF_1 Letter Amino acid
22 19 6 D Aspartic acid
13 4 0 C Cysteine
14 22 10 E Glutamic acid
15 19 0 H Histidine
22 20 3 I Isoleucine
33 29 3 A Alanine
19 20 12 R Arginine
9 11 7 N Asparagine
20 20 3 T Threonine
6 2 0 W Tryptophan
22 26 6 V Valine
16 44 10 L Leucine
11 10 10 F Phenylalanine
21 16 6 P Proline
14 21 4 Q Glutamine
8 7 0 M Methionine
10 4 7 Y Tyrosine
28 30 6 G Glycine
19 22 4 S Serine
13 18 3 K Lycine 

3USC_1|Chains A[auth S], C[auth T]|Hydrogenase-1 small chain|Escherichia coli (562)
>8PND_1|Chains A, B|Porphobilinogen deaminase|Homo sapiens (9606)
>1EOF_1|Chain A|POTASSIUM CHANNEL KV1.1|Aplysia californica (6500)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
3USC , Knot 143 335 0.82 40 204 319 
LENKPRIPVVWIHGLECTCCTESFIRSAHPLAKDVILSLISLDYDDTLMAAAGTQAEEVFEDIITQYNGKYILAVEGNPPLGEQGMFCISSGRPFIEKLKRAAAGASAIIAWGTCASWGCVQAARPNPTQATPIDKVITDKPIIKVPGCPPIPDVMSAIITYMVTFDRLPDVDRMGRPLMFYGQRIHDKCYRRAHFDAGEFVQSWDDDAARKGYCLYKMGCKGPTTYNACSSTRWNDGVSFPIQSGHGCLGCAENGFWDRGSFYSRVVDIPQMGTHSTADTVGLTALGVVAAAVGVHAVASAVDQRRRHNQQPTETEHQPGNEDKQARSHHHHHH
8PND , Knot 154 364 0.83 40 209 341 
MGSSHHHHHHSSGENLYFQGMRVIRVGTRKSQLARIQTDSVVATLKASYPGLQFEIIAMSTTGDKILDTALSKIGEKSLFTKELEHALEKNEVDLVVHSLKDLPTVLPPGFTIGAICKRENPHDAVVFHPKFVGKTLETLPEKSVVGTSSLRRAAQLQRKFPHLEFRSIQGNLNTRLRKLDEQQEFSAIILATAGLQRMGWHNRVGQILHPEECMYAVGQGALGVEVRAKDQDILDLVGVLHDPETLLRCIAERAFLRHLEGGCSVPVAVHTAMKDGQLYLTGGVWSLDGSDSIQETMQATIHVPAQHEDGPEDDPQLVGITARNIPRGPQLAAQNLGISLANLLLSKGAKNILDVARQLNDAH
1EOF , Knot 51 100 0.78 32 83 97 
ERVVINVSGLRFETQLKTLNQFPDTLLGNPQKRNRYYDPLRNEYFFDRNRPSFDAILYFYQSGGRLRRPVAVPLDVFSEEIKFYELGENAFERYREDEGF

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(3USC_1)}(2) \setminus P_{f(8PND_1)}(2)|=80\), \(|P_{f(8PND_1)}(2) \setminus P_{f(3USC_1)}(2)|=85\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:10001011111101100000000110010111001110110100000111111001001100110000100111101011110011101001011100100111110111111001011010110101001011001100011101110111101101110011010011010011011110100100000001010110110010001100100100110011000010000010011011100101011010011100101000110110110000100111011111111111011101100000000010000001100000100000000
Pair \(Z_2\) Length of longest common subsequence
3USC_1,8PND_1 165 7
3USC_1,1EOF_1 183 3
8PND_1,1EOF_1 182 4

Newick tree

 
[
	1EOF_1:93.98,
	[
		3USC_1:82.5,8PND_1:82.5
	]:11.48
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{699 }{\log_{20} 699}-\frac{335}{\log_{20}335})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 3USC_1 8PND_1 124 121
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]