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Parikh vectors
5JRI_1 2AVW_1 5DGG_1 Letter Amino acid
4 0 3 C Cysteine
14 17 11 E Glutamic acid
34 24 16 G Glycine
12 11 7 H Histidine
18 28 8 K Lycine
8 10 12 P Proline
7 6 1 W Tryptophan
26 18 13 V Valine
27 20 11 A Alanine
13 7 10 R Arginine
10 12 6 Q Glutamine
12 20 9 S Serine
12 22 9 N Asparagine
19 17 8 I Isoleucine
34 22 17 L Leucine
11 5 7 M Methionine
26 23 18 D Aspartic acid
7 18 4 F Phenylalanine
22 20 11 T Threonine
14 11 9 Y Tyrosine 

5JRI_1|Chains A, B|Pyridine nucleotide-disulfide oxidoreductase|Synechocystis sp. PCC 6803 (1148)
>2AVW_1|Chains A, B, C, D, E, F|IgG-degrading protease|Streptococcus pyogenes (293653)
>5DGG_1|Chains A, B|Uncharacterized protein|Legionella pneumophila subsp. pneumophila (272624)
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 \(w=f(c)\)
5JRI , Knot 145 330 0.85 40 208 322 
GSHMVKLSSKNLDARLDTVYDAIVLGGGMGGLSAAIYLARYGLKCLVVEKGRGRSFWMQDLRNYVGLDPDTPGRDIITHSTQQALHWGADLLRGYVEDVTDEGDTLAVKVKVGKKDSLYPIFRTKYVIAATGIIDNLPQLEDMQNVYDYAGYTLHVCMICDGFDMWDQKAVLIAGTEGQINAAFVLNWFTPYITVLTHGLCTVGDEMKAKLADHGYPLHEAAITKFLGEDHKMSGVELVDGTVMEATTGLINMGSVYHNHYLKGIEGLEWDGENLVTNDMAQTSHPRIFALGDLKKGLNQVSVAVADGTLAATQIWRNIRRASEPRKWIH
2AVW , Knot 139 311 0.85 38 195 296 
MDSFSANQEIRYSEVTPYHVTSVWTKGVTPPANFTQGEDVFHAPYVANQGWYDITKTFNGKDDLLAGAATAGNMLHWWFDQNKDQIKRYLEEHPEKQKINFNGEQMFDVKEAIDTKNHQLDSKLFEYFKEKAFPYLSTKHLGVFPDHVIDMFINGYRLSLTNHGPTPVKEGSKDPRGGIFDAVFTRGDQSKLLTSRHDFKEKNLKEISDLIKKELTEGKALGLSHTYANVRINHVINLWGADFDSNGNLKAIYVTDSDSNASIGMKKYFVGVNSAGKVAISAKEIKEDNIGAQVLGLFTLSTGQDSWNQTN
5DGG , Knot 91 190 0.83 40 147 183 
TERDQMQDHDMTLLMPKSQGRIVVMAVLNRYDSHSANAIIETLASDVFNPEVHYIMIPVGPGHWRGVYLSKPQGGTSDTAYDLELFDPYGPEGAAVLDDYVLDLLNQCGVPKELVNIRHTGPKHPQGDAYSCGDFTCAYSHKKMKEFGAPEGSYNPILIDTLDNLGNEDNVLRMTTREETRALVDKDRGR

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 \(c\).

\(|P_{f(5JRI_1)}(2) \setminus P_{f(2AVW_1)}(2)|=86\), \(|P_{f(2AVW_1)}(2) \setminus P_{f(5JRI_1)}(2)|=73\). 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 1:100110100001010100100111111111110111011001100111001010011100100011101001100110000001101110110101001000100111010110000101110000111101110011010010010001100101011001101100011111100101011111011010101100110011001010110010110011100111000010110110101101001110110100000101101101010011000110000101111101001100101111010111001100100100100110
Pair \(Z_2\) Length of longest common subsequence
5JRI_1,2AVW_1 159 4
5JRI_1,5DGG_1 175 3
2AVW_1,5DGG_1 170 3

Newick tree

 
[
	5DGG_1:88.39,
	[
		5JRI_1:79.5,2AVW_1:79.5
	]:8.89
]

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 \((0.85)(0.8)(\frac{641 }{\log_{20} 641}-\frac{311}{\log_{20}311})=91.6\)
Status Protein1 Protein2 d d1/2
Query variables 5JRI_1 2AVW_1 121 115.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]