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Parikh vectors
5YLT_1 7HYY_1 4ZOA_1 Letter Amino acid
5 5 24 Q Glutamine
9 3 26 F Phenylalanine
19 14 62 V Valine
16 10 62 A Alanine
11 5 34 N Asparagine
24 14 43 S Serine
9 6 18 H Histidine
12 5 25 I Isoleucine
24 7 75 E Glutamic acid
24 14 57 G Glycine
16 12 82 L Leucine
16 4 30 P Proline
3 3 6 W Tryptophan
15 8 23 Y Tyrosine
9 9 25 R Arginine
14 9 33 D Aspartic acid
6 2 20 M Methionine
14 6 41 T Threonine
5 6 4 C Cysteine
12 2 41 K Lycine 

5YLT_1|Chain A|Histone-lysine N-methyltransferase SETD7|Homo sapiens (9606)
>7HYY_1|Chains A, B[auth C]|Protease 2A|Coxsackievirus A16 (31704)
>4ZOA_1|Chains A, B|Lin1840 protein|Listeria innocua serovar 6a (strain CLIP 11262) (272626)
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 \(w=f(c)\)
5YLT , Knot 119 263 0.84 40 179 255 
GSSGSSGKDNIRHGVCWIYYPDGGSLVGEVNEDGEMTGEKIAYVYPDERTALYGKFIDGEMIEGKLATLMSTEEGRPHFELMPGNSVYHFDKSTSSCISTNALLPDPYESERVYVAESLISSAGEGLFSKVAVGPNTVMSFYNGVRITHQEVDSRDWALNGNTLSLDEETVIDVPEPYNHVSKYCASLGHKANHSFTPNCIYDMFVHPRFGPIKCIRTLRAVEADEELTVAYGYDHSPPGKSGPEAPEWYQVELKAFQATQQK
7HYY , Knot 70 144 0.80 40 112 139 
SGAIYVGNYRVVNRHLATHNDWANLVWEDSSRDLLVSSTTAQGCDTIARCDCQTGVYYCSSRRKHYPVSFSKPSLIFVEASEYYPARYQSHLMLAVGHSEPGDCGGILRCQHGVVGIVSTGGNGLVGFADVRDLLWLDEEAMEQ
4ZOA , Knot 275 731 0.82 40 266 655 
MEQEKVQELVSQMTLDEKIAQCLQLSPFLFKGTNKNAELTGPLLQEMKLTDAHTENAGSVLGSSSALDMIGIQEAYLKTNRLGIPLVFMADVIHGYKTVFPIPLALGCSFDRETVRVMAEVSALEATADGHHVTFSPMLDLVRDPRWGRVMESTGEDPFLNSELGKAMVDGYQGDASKLNENLEQMAACVKHFAAYGAAEAGLEYNTVNMSTRELYQNYLPAYNAAIQAGAKLVMTAFNVVDGIPATMNKWLNRDVLRGEMEFDGVLISDWGAVAEVINHGTARNPKEAAQFSMEAGVDLEMMTTCYIHELKGLIEEGKLSENLLDEAVLRMLNLKNDLGLFEDPYRGLKNNDRTKDILTDESRGKARAAGVESAVLLENKSRLLPLAKEAKIALVGPLATSPDILGGWNVYGEEKDGINVETGLREVFETVEVVSTEYTELSEEDKVAVKAAVQNMDVVVLALGEKNEWGGEAGSLATIRLPEAQYQLAKFVQTLGKPVVITLFNGRPLEVKELAESSDALLELWFPGTEAGRVTADLLSGASNPSGKLSMSFPQTTGQIPVYYNHLRTGRPQTPENKGERYVSHYLDIPNEPFYPFGYGKSYSEFELKTSSLPKELNLGESLHVEVTIKNISDIAGKEVIQVYLQDVTASISRPVKELKAFEKVALQAGEEKTVTFELTSEAFSFYNHQLEKVQEPGLHRVFVGTSSEDVDVFEVEVGGYVLEHHHHHH

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 \(c\).

\(|P_{f(5YLT_1)}(2) \setminus P_{f(7HYY_1)}(2)|=120\), \(|P_{f(7HYY_1)}(2) \setminus P_{f(5YLT_1)}(2)|=53\). 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 1:10010010001001101100101101110100010101001101010000110101101011010110110000101010111100100100000001000111101000001011001100110111001111100110100110100001000011101001010000110110100010000101100100010100100111010111100100101101000101101000011100110110100101011010000
Pair \(Z_2\) Length of longest common subsequence
5YLT_1,7HYY_1 173 4
5YLT_1,4ZOA_1 173 5
7HYY_1,4ZOA_1 206 3

Newick tree

 
[
	4ZOA_1:97.80,
	[
		5YLT_1:86.5,7HYY_1:86.5
	]:11.30
]

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 \((0.85)(0.8)(\frac{407 }{\log_{20} 407}-\frac{144}{\log_{20}144})=78.9\)
Status Protein1 Protein2 d d1/2
Query variables 5YLT_1 7HYY_1 99 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]