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Parikh vectors
4HKN_1 2PTT_1 8PSU_1 Letter Amino acid
7 4 31 A Alanine
7 1 16 Q Glutamine
13 10 32 E Glutamic acid
10 9 24 K Lycine
9 6 26 V Valine
11 9 35 R Arginine
7 4 24 D Aspartic acid
20 6 23 G Glycine
12 9 43 L Leucine
5 0 13 C Cysteine
11 9 17 I Isoleucine
6 2 6 M Methionine
11 3 18 F Phenylalanine
16 5 31 S Serine
11 9 8 N Asparagine
16 3 13 H Histidine
3 4 22 P Proline
7 9 18 T Threonine
1 1 4 W Tryptophan
8 7 15 Y Tyrosine 

4HKN_1|Chain A|Tankyrase-2|Homo sapiens (9606)
>2PTT_1|Chain A|CD48 antigen|Mus musculus (10090)
>8PSU_1|Chain A|Polymerase acidic protein (PA-like)|Tilapia lake virus (1549864)
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 \(w=f(c)\)
4HKN , Knot 91 191 0.83 40 146 184 
MHHHHHHSSGVDLGTENLYFQSMLNTSGSGTILIDLSPDDKEFQSVEEEMQSTVREHRDGGHAGGIFNRYNILKIQKVCNKKLWERYTHRRKEVSEENHNHANERMLFHGSPFVNAIIHKGFDERHAYIGGMFGAGIYFAENSSKSNQYVYGIGGGTGCPVHKDRSCYICHRQLLFCRVTLGKSFLQFSAM
2PTT , Knot 55 110 0.78 38 90 106 
MARIRARGSIPDINAYTGSNVTLKIHKDPLGPYRRITWLHTKNQKILEYNYNSTKTIFESEFKGRVYLEENNGALHISNVRKEDKGTYYMRVLRETENELKITLEVFDPV
8PSU , Knot 178 419 0.85 40 239 402 
MDSRFAQLTGVFCDDFTYSEGSRRFLSSYSTVERRPGVPVEGDCYDCLKNKWIAFELEGQPRKFPKATVRCILNNDATYVCSEQEYQQICKVQFKDYLEIDGVVKVGHKASYDAELRERLLELPHPKSGPKPRIEWVAPPRLADISKETAELKRQYGFFECSKFLACGEECGLDQEARELILNEYARDREFEFRNGGWIQRYTVASHKPATQKILPLPASAPLARELLMLIARSTTQAGKVLHSDNTSILAVPVMRDSGKHSKRRPTASTHHLVVGLSKPGCEHDFEFDGYRAAVHVMHLDPKQSANIGEQDFVSTREIYKLDMLELPPISRKGDLDRASGLETRWDVILLLECLDSTRVSQAVAQHFNRHRLALSVCKDEFRKGYQLASEIRGTIPLSSLYYSLCAVRLRMTVHPFAR

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 \(c\).

\(|P_{f(4HKN_1)}(2) \setminus P_{f(2PTT_1)}(2)|=103\), \(|P_{f(2PTT_1)}(2) \setminus P_{f(4HKN_1)}(2)|=47\). 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 1:10000000011011000101001100010101110101000010010001000100000110111110000110100100001100000000010000000100011101011101110011000010111111111011000000000101111101011000000010000111001011001101011
Pair \(Z_2\) Length of longest common subsequence
4HKN_1,2PTT_1 150 3
4HKN_1,8PSU_1 197 3
2PTT_1,8PSU_1 217 4

Newick tree

 
[
	8PSU_1:11.54,
	[
		4HKN_1:75,2PTT_1:75
	]:36.54
]

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 \((0.85)(0.8)(\frac{301 }{\log_{20} 301}-\frac{110}{\log_{20}110})=59.7\)
Status Protein1 Protein2 d d1/2
Query variables 4HKN_1 2PTT_1 74 58
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]