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Parikh vectors
9VVS_1 6ISB_1 2IFX_1 Letter Amino acid
11 12 5 I Isoleucine
5 7 4 M Methionine
4 12 0 N Asparagine
21 8 9 D Aspartic acid
41 11 8 G Glycine
22 22 4 S Serine
5 6 2 W Tryptophan
11 13 8 E Glutamic acid
13 5 6 H Histidine
41 13 8 L Leucine
6 11 5 Y Tyrosine
22 23 7 V Valine
49 11 12 A Alanine
9 15 2 Q Glutamine
9 12 7 P Proline
3 10 7 F Phenylalanine
14 18 6 T Threonine
18 10 6 R Arginine
1 6 2 C Cysteine
4 7 6 K Lycine 

9VVS_1|Chains A, B|Dehydrogenase|Kutzneria albida DSM 43870 (1449976)
>6ISB_1|Chains A, B, C|CD226 antigen|Homo sapiens (9606)
>2IFX_1|Chains A, B|Hypothetical protein|Cupriavidus necator (106590)
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 \(w=f(c)\)
9VVS , Knot 119 309 0.73 40 143 270 
MGSSHHHHHHSSGLVPRGSHMDKRERVTVLGLGRMGAALAGALLAAGHPTTVWNRSAAKAGDLVARGAVLAGTAAEAVRASEVVVVCVTDYDVAGDLLGPLAGELAGRALINLTSGSPEQARRSADWAAEHGIDYLDGTIMTQPQGIGQPGSVLLLSGSAEVHSGRKALIELWGDGDYLGADAGLSALYDLALLGVMWSTIASFLHGAALLGTEDVTAQQFLPKVERWLGAVASFLPGIAAQADSGDYATDHGGLDINVAGLDLLARVSRAQGIGTDVPAALHELYARRLAQGSANDGLASLIDTIRRG
6ISB , Knot 109 232 0.85 40 171 225 
EEVLWHTSVPFAENMSLECVYPSMGILTQVEWFKIGTQQDSIAIFSPTHGMVIRKPYAERVYFLNSTMASNNMTLFFRNASEDDVGYYSCSLYTYPQGTWQKVIQVVQSDSFEAAVPSNSHIVSEPGKNVTLTCQPQMTWPVQAVRWEKIQPRQIDLLTYCNLVHGRNFTSKFPRQIVSNCSHGRWSVIVIPDVTVSDSGLYRCYLQASAGENETFVMRLTVAEGKTDNQYT
2IFX , Knot 59 114 0.81 38 87 112 
GMIRLLYLLVKPAGMSDETFRAECLRHYEMSHDVPGLHKYEVRLVAEQPTDTHVPFFDIGHVDAIGECWFKDDAAYATYMASDIRKAWFEHGKTFIGQLKPFRTAPVAGDEPAS

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 \(c\).

\(|P_{f(9VVS_1)}(2) \setminus P_{f(6ISB_1)}(2)|=71\), \(|P_{f(6ISB_1)}(2) \setminus P_{f(9VVS_1)}(2)|=99\). 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 1:110000000000111101001000001011111011111111111110100110001101101110111111011011010011110100001110111111101110111010010100100010111001100101011001011101101111010101001001110111010011101110110011111111001101101111110001010011101001111110111111101001001000111010111101110100101110011111001010011010100111011001001
Pair \(Z_2\) Length of longest common subsequence
9VVS_1,6ISB_1 170 3
9VVS_1,2IFX_1 150 4
6ISB_1,2IFX_1 162 3

Newick tree

 
[
	6ISB_1:85.53,
	[
		9VVS_1:75,2IFX_1:75
	]:10.53
]

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 \((0.85)(0.8)(\frac{541 }{\log_{20} 541}-\frac{232}{\log_{20}232})=88.3\)
Status Protein1 Protein2 d d1/2
Query variables 9VVS_1 6ISB_1 104 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]