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Parikh vectors
8ZFA_1 5JJX_1 7KIX_1 Letter Amino acid
4 9 0 W Tryptophan
30 23 7 A Alanine
31 21 4 R Arginine
16 9 6 N Asparagine
16 14 8 G Glycine
6 7 10 H Histidine
25 18 3 I Isoleucine
20 14 4 S Serine
16 19 4 Y Tyrosine
21 18 5 V Valine
19 15 3 Q Glutamine
22 43 9 E Glutamic acid
27 40 5 L Leucine
18 8 4 F Phenylalanine
18 5 8 T Threonine
28 15 4 D Aspartic acid
7 4 0 C Cysteine
22 15 2 K Lycine
7 7 3 M Methionine
8 10 2 P Proline 

8ZFA_1|Chain A|Guanine nucleotide-binding protein G(s) subunit alpha isoforms short|Homo sapiens (9606)
>5JJX_1|Chain A|Squamous cell carcinoma antigen recognized by T-cells 3|Homo sapiens (9606)
>7KIX_1|Chain A|Anti-CRISPR protein AcrIE2|Pseudomonas phage JBD16C (1777051)
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 \(w=f(c)\)
8ZFA , Knot 157 361 0.85 40 219 345 
MGCTLSAEDKAAVERSKMIEKQLQKDKQVYRATHRLLLLGADNSGKSTIVKQMRIYHVNGYSEEECKQYKAVVYSNTIQSIIAIIRAMGRLKIDFGDSARADDARQLFVLAGAAEEGFMTAELAGVIKRLWKDSGVQACFNRSREYQLNDSAAYYLNDLDRIAQPNYIPTQQDVLRTRVKTSGIFETKFQVDKVNFHMFDVGAQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCSVDTENARRIFNDCRDIIQRMHLRQYELL
5JJX , Knot 132 314 0.80 40 169 291 
GGEYEWEYDEEEEKNQLEIERLEEQLSINVYDYNCHVDLIRLLRLEGELTKVRMARQKMSEIFPLTEELWLEWLHDEISMAQDGLDREHVYDLFEKAVKDYICPNIWLEYGQYSVGGIGQKGGLEKVRSVFERALSSVGLHMTKGLALWEAYREFESAIVEAARLEKVHSLFRRQLAIPLYDMEATFAEYEEWSEDPIPESVIQNYNKALQQLEKYKPYEEALLQAEAPRLAEYQAYIDFEMKIGDPARIQLIFERALVENCLVPDLWIRYSQYLDRQLKVKDLVLSVHNRAIRNCPWTVALWSRYLLAMERHG
7KIX , Knot 49 91 0.81 36 77 86 
MNTYLIDPRKNNDNSGERFTVDAVDITAAAKSAAQQILGEEFEGLVYRETGESNGSGMFQAYHHLHGTNRTETTVGYPFHVMELEHHHHHH

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 \(c\).

\(|P_{f(8ZFA_1)}(2) \setminus P_{f(5JJX_1)}(2)|=112\), \(|P_{f(5JJX_1)}(2) \setminus P_{f(8ZFA_1)}(2)|=62\). 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 1:1100101000111000011000100000100100011111100010001100101001010000000000111000010011111011101010110010100100111111110011101011111001100011010100000001000110010010011010011000011000100011100010100101011011100000001100100101111110000000100110010011000110010111110000111001111000100011011000010010101100101001001100011010010101000001010001000010011000001100101000011
Pair \(Z_2\) Length of longest common subsequence
8ZFA_1,5JJX_1 174 3
8ZFA_1,7KIX_1 190 4
5JJX_1,7KIX_1 172 3

Newick tree

 
[
	8ZFA_1:92.72,
	[
		5JJX_1:86,7KIX_1:86
	]:6.72
]

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 \((0.85)(0.8)(\frac{675 }{\log_{20} 675}-\frac{314}{\log_{20}314})=99.8\)
Status Protein1 Protein2 d d1/2
Query variables 8ZFA_1 5JJX_1 129 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]