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Parikh vectors
8FGZ_1 9GSO_1 5UMY_1 Letter Amino acid
10 10 14 G Glycine
12 10 11 P Proline
9 10 1 Y Tyrosine
14 16 7 R Arginine
16 4 2 I Isoleucine
16 7 5 F Phenylalanine
18 12 11 S Serine
6 9 5 T Threonine
9 2 9 H Histidine
11 11 13 D Aspartic acid
15 12 7 Q Glutamine
14 8 3 E Glutamic acid
13 8 3 M Methionine
13 9 11 A Alanine
6 4 1 C Cysteine
28 16 13 L Leucine
14 11 4 K Lycine
4 3 3 W Tryptophan
19 12 14 V Valine
11 3 3 N Asparagine 

8FGZ_1|Chain A|Androgen receptor|Homo sapiens (9606)
>9GSO_1|Chain A|DNA binding domain of J-DNA Binding Protein 3 (JBP3)|Leishmania tarentolae (5689)
>5UMY_1|Chains A, B|Glyoxalase/bleomycin resisance protein/dioxygenase|Streptomyces sp. CB03234 (1703937)
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 \(w=f(c)\)
8FGZ , Knot 121 258 0.86 40 189 249 
SHIEGYECQPIFLNVLEAIEPGVVCAGHDNNQPDSFAALHSSLNELGERQLVHVVKWAKALPGFRNLHVDDQMAVIQYSWMGLMVFAMGWRSFTNVNSRMLYFAPDLVFNEYRMHKSRMYSQCVRMRHLSQEFGWLQITPQEFLCMKALLLFSIIPVDGLKNQKFFDELRMNYIKELDRIIACKRKNPTSCSRRFYQLTKLLDSVQPIARELYQLTFDLLIKSHMVSVDFPEMMAEIISVQVPKILSGKVKPIYFHTQ
9GSO , Knot 86 177 0.83 40 138 175 
DKEYIPLPPLRDMQDMSKVLFLLSTDKKRYPDGRHRTLDYFRASVEMFVTEVRQEYKRQYQQAQRGGRAMQRFTWKNSGELAICFACCCDNVKLLYDSLQPGPLKPLWDAFVSQLAPMLIIQSRVPEMMLSSQTYHTKYMDWVKGGTRYGGEQSTANVRFPSVADRRVKVETYLRSG
5UMY , Knot 68 140 0.80 40 105 135 
HHHHHHSSGLVPRGSHMAISHVQLFSVPVSDQEKAKDFYVETVGFDLLADQPGVHGRWLQVAPKGADTSLVLVDWFPTMPPGSLRGLLLRTDDVDADCARLQERGVAVDGPKNTPWGRQAMFSDPDGNVIGLNQPSASAG

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 \(c\).

\(|P_{f(8FGZ_1)}(2) \setminus P_{f(9GSO_1)}(2)|=121\), \(|P_{f(9GSO_1)}(2) \setminus P_{f(8FGZ_1)}(2)|=70\). 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 1:001010000111101101101111011000001001111000100110001101101101111100101000111100011111111111001001000110111011100001000010000101001000111101010011010111110111101100001100101001001001110000010000001001001100101110010010101110001101011011101101011011010101101000
Pair \(Z_2\) Length of longest common subsequence
8FGZ_1,9GSO_1 191 4
8FGZ_1,5UMY_1 172 4
9GSO_1,5UMY_1 159 3

Newick tree

 
[
	8FGZ_1:94.36,
	[
		5UMY_1:79.5,9GSO_1:79.5
	]:14.86
]

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 \((0.85)(0.8)(\frac{435 }{\log_{20} 435}-\frac{177}{\log_{20}177})=76.1\)
Status Protein1 Protein2 d d1/2
Query variables 8FGZ_1 9GSO_1 103 85
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]