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Parikh vectors
6HAU_1 5HRJ_1 7KDV_1 Letter Amino acid
11 8 7 C Cysteine
7 11 16 H Histidine
19 0 29 K Lycine
20 11 34 S Serine
19 4 42 T Threonine
19 5 31 A Alanine
12 0 33 N Asparagine
10 7 34 D Aspartic acid
17 5 45 P Proline
12 0 32 Y Tyrosine
16 12 39 V Valine
13 4 23 Q Glutamine
7 14 51 G Glycine
37 8 64 L Leucine
7 0 13 M Methionine
2 2 13 W Tryptophan
8 5 29 R Arginine
9 8 26 E Glutamic acid
19 2 32 I Isoleucine
9 3 37 F Phenylalanine 

6HAU_1|Chains A, B|mRNA export factor ICP27 homolog|Saimiriine herpesvirus 2 (strain 11) (10383)
>5HRJ_1|Chain A|Scavenger receptor cysteine-rich type 1 protein M130|Sus scrofa (9823)
>7KDV_1|Chains A, C, E, G, I, K|Beta-galactosidase|Mus musculus (10090)
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 \(w=f(c)\)
6HAU , Knot 125 273 0.85 40 182 260 
GPLLKLFDISILPKSGEPKLFLPVPSLPCQEAEKTNDKYVLAMAQRAMHDVPISSKQLTANLLPVKFKPLLSIVRYTPNYYYWVSMRKETIASANLCTVAAFLDESLCWGQQYLKNDFIFSENGKDIILDTSSALLSQLVHKIKMLPFCHCLMQTTPQDHIVKQVCYLIASNNRILDAVRYLQTSVIKSPIVLLLAYAVCLPAAIICTKNETQLYSHCMRILKEYRPGDVMNILHESLTQHLNKCPSSTCAYTTRAIVGTKANTTGLFFLPTQ
5HRJ , Knot 53 109 0.76 32 85 103 
PRLVGGDIPCSGRVEVQHGDTWGTVCDSDFSLEAASVLCRELQCGTVVSLLGGAHFGEGSGQIWAEEFQCEGHESHLSLCPVAPRPDGTCSHSRDVGVVCSVDHHHHHH
7KDV , Knot 247 630 0.84 40 277 593 
DRHHHHHHGSVTQRTFKLDYSRDRFLKDGQPFRYISGSIHYFRIPRFYWEDRLLKMKMAGLNAIQMYVPWNFHEPQPGQYEFSGDRDVEHFIQLAHELGLLVILRPGPYICAEWDMGGLPAWLLEKQSIVLRSSDPDYLVAVDKWLAVLLPKMKPLLYQNGGPIITVQVENEYGSYFACDYDYLRFLVHRFRYHLGNDVILFTTDGASEKMLKCGTLQDLYATVDFGTGNNITQAFLVQRKFEPKGPLINSEFYTGWLDHWGKPHSTVKTKTLATSLYNLLARGANVNLYMFIGGTNFAYWNGANTPYEPQPTSYDYDAPLSEAGDLTKKYFALREVIQMFKEVPEGPIPPSTPKFAYGKVALRKFKTVAEALGILCPNGPVKSLYPLTFTQVKQYFGYVLYRTTLPQDCSNPKPIFSSPFNGVRDRAYVSVDGVPQGILDRNLMTALNIQGKAGATLDILVENMGRVNYGRFINDFKGLISNMTINSTVLTNWTVFPLDTEAMVRNHLWGREASDGGHLDGRSTSNSSDLILPTFYVGNFSIPSGIPDLPQDTFIQFPGWSKGQVWINGFNLGRYWPTMGPQKTLFVPRNILTTSAPNNITVLELEFAPCSEGTPELCTVEFVDTPVIS

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 \(c\).

\(|P_{f(6HAU_1)}(2) \setminus P_{f(5HRJ_1)}(2)|=143\), \(|P_{f(5HRJ_1)}(2) \setminus P_{f(6HAU_1)}(2)|=46\). 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 1:111101101011100101011111101100010000000111110011001110000101011110101110110001000011010000110101001111100010110001000111000100111000011100110010111100011000100011001001110000110110010001100111111101101111110000000100001011000011011011000100010001000010000111100100011111100
Pair \(Z_2\) Length of longest common subsequence
6HAU_1,5HRJ_1 189 3
6HAU_1,7KDV_1 181 4
5HRJ_1,7KDV_1 228 6

Newick tree

 
[
	5HRJ_1:10.02,
	[
		6HAU_1:90.5,7KDV_1:90.5
	]:18.52
]

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 \((0.85)(0.8)(\frac{382 }{\log_{20} 382}-\frac{109}{\log_{20}109})=83.5\)
Status Protein1 Protein2 d d1/2
Query variables 6HAU_1 5HRJ_1 112 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]