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Parikh vectors
1SPR_1 1ZDN_1 8TJK_1 Letter Amino acid
7 13 32 G Glycine
1 1 4 W Tryptophan
5 10 6 N Asparagine
4 6 20 D Aspartic acid
4 1 8 Q Glutamine
8 10 20 T Threonine
8 7 29 R Arginine
7 8 6 K Lycine
0 2 9 M Methionine
5 5 13 F Phenylalanine
6 5 10 Y Tyrosine
5 10 43 A Alanine
3 2 2 C Cysteine
7 15 27 E Glutamic acid
3 5 9 H Histidine
3 10 12 I Isoleucine
11 22 47 L Leucine
3 13 17 P Proline
9 4 16 S Serine
5 9 14 V Valine 

1SPR_1|Chains A, B, C, D|SRC TYROSINE KINASE SH2 DOMAIN|Rous sarcoma virus (11886)
>1ZDN_1|Chains A, B|Ubiquitin-conjugating enzyme E2S|Homo sapiens (9606)
>8TJK_1|Chains A, B|RedM|uncultured bacterium (77133)
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 \(w=f(c)\)
1SPR , Knot 56 104 0.83 38 88 100 
QAEEWYFGKITRRESERLLLNPENPRGTFLVRESETTKGAYCLSVSDFDNAKGLNVKHYKIRKLDSGGFYITSRTQFSSLQQLVAYYSKHADGLCHRLTNVCPT
1ZDN , Knot 77 158 0.82 40 117 151 
GSMNSNVENLPPHIIRLVYKEVTTLTADPPDGIKVFPNEEDLTDLQVTIEGPEGTPYAGGLFRMKLLLGKDFPASPPKGYFLTKIFHPNVGANGEICVNVLKRDWTAELGIRHVLLTIKCLLIHPNPESALNEEAGRLLLENYEEYAARARLLTEIHG
8TJK , Knot 145 344 0.82 40 183 318 
MSDTSPGPEHAPAIDRLLQIATGFMASKVLLVAASLGLFTELAAGPLRGEELRARLRLHPRSARDFFDTLVALGVLERTNGAYANTPATAQYLVRGKSAYLGGLLEMSDARMYELWGRLDEGLRTGNPQNEIRTGEEGIYATLYDDPDRLDAFQQAMTGLSMRSAHALAEAIDWSAYRTVADIGCAEGTVLIHLLERHPHLRGTGFDLAAVRPSFQRRHEESGLGDRLAFRAGDFFAEPLPQADALVFGHILSNWALPKAKTLLRKAHEALPEGGIVVIYETLIDDERRENVPGLLMSLTMLLETPGGFEYTGADCREWLADAGFRESRVQYLAGPESMVIATK

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 \(c\).

\(|P_{f(1SPR_1)}(2) \setminus P_{f(1ZDN_1)}(2)|=57\), \(|P_{f(1ZDN_1)}(2) \setminus P_{f(1SPR_1)}(2)|=86\). 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 1:01001011010000000111010010101110000000110010100100101101000010010011101000001001001110000010110001001010
Pair \(Z_2\) Length of longest common subsequence
1SPR_1,1ZDN_1 143 4
1SPR_1,8TJK_1 173 3
1ZDN_1,8TJK_1 158 4

Newick tree

 
[
	8TJK_1:86.28,
	[
		1SPR_1:71.5,1ZDN_1:71.5
	]:14.78
]

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 \((0.85)(0.8)(\frac{262 }{\log_{20} 262}-\frac{104}{\log_{20}104})=50.2\)
Status Protein1 Protein2 d d1/2
Query variables 1SPR_1 1ZDN_1 67 53.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]