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Parikh vectors
1TOO_1 4JTV_1 6OCC_1 Letter Amino acid
11 20 15 V Valine
9 21 19 N Asparagine
2 9 7 C Cysteine
8 18 15 P Proline
6 24 11 T Threonine
14 30 16 S Serine
2 6 4 W Tryptophan
5 17 15 A Alanine
11 16 21 E Glutamic acid
1 10 15 H Histidine
15 24 34 L Leucine
3 11 13 R Arginine
5 20 19 I Isoleucine
4 14 6 Y Tyrosine
6 2 4 M Methionine
8 10 14 F Phenylalanine
8 15 16 D Aspartic acid
12 7 8 Q Glutamine
8 22 15 G Glycine
15 25 22 K Lycine 

1TOO_1|Chain A|Interleukin-1 beta|Homo sapiens (9606)
>4JTV_1|Chains A, C, E, G, I, K|Hemagglutinin|Influenza A virus (641501)
>6OCC_1|Chain A|Tyrosine-protein kinase JAK2|Homo sapiens (9606)
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 \(w=f(c)\)
1TOO , Knot 76 153 0.83 40 120 150 
APVRSLNCTLRDSQQKSLVMSGPYELKALHLQGQDMEQQVVFSMSFVQGEESNDKIPVALGLKEKNLYLSCVLKDDKPTLQLESVDPKNYPKKKMEKRFVFNKIEINNKLEFESAQFPNWYISTSQAENMPVFLGGTKGGQDITDWTMQFVSS
4JTV , Knot 145 321 0.87 40 207 312 
DTLCIGYHANNSTDTVDTVLEKNVTVTHSVNLLEDKHNGKLCKLRGVAPLHLGKCNIAGWILGNPECESLSTASSWSYIVETPSSDNGTCYPGDFIDYEELREQLSSVSSFERFEIFPKTSSWPNHDSNKGVTAACPHAGAKSFYKNLIWLVKKGNSYPKLSKSYINDKGKEVLVLWGIHHPSTSADQQSLYQNADTYVFVGSSRYSKKFKPEIAIRPKVRDQEGRMNYYWTLVEPGDKITFEATGNLVVPRYAFAMERNAGSGIIISDTPVHDCNTTCQTPKGAINTSLPFQNIHPITIGKCPKYVKSTKLRLATGLRNI
6OCC , Knot 127 289 0.83 40 188 280 
VFHKIRNEDLIFNESLGQGTFTKIFKGVRREVGDYGQLHETEVLLKVLDKAHRNYSESFFEAASMMSKLSHKHLVLNYGVCVCGDENILVQEFVKFGSLDTYLKKNKNCINILWKLEVAKQLAAAMHFLEENTLIHGNVCAKNILLIREEDRKTGNPPFIKLSDPGISITVLPKDILQERIPWVPPECIENPKNLNLATDKWSFGTTLWEICSGGDKPLSALDSQRKLQFYEDRHQLPAPKAAELANLINNCMDYEPDHRPSFRAIIRDLNSLFTPDLVPRGSHHHHHH

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 \(c\).

\(|P_{f(1TOO_1)}(2) \setminus P_{f(4JTV_1)}(2)|=51\), \(|P_{f(4JTV_1)}(2) \setminus P_{f(1TOO_1)}(2)|=138\). 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 1:111001000100000001110110010110101001000111010110100000011111110000101001100001010100101000100010001110010100010100101101010000100111111100110010010101100
Pair \(Z_2\) Length of longest common subsequence
1TOO_1,4JTV_1 189 3
1TOO_1,6OCC_1 168 3
4JTV_1,6OCC_1 169 3

Newick tree

 
[
	4JTV_1:91.44,
	[
		1TOO_1:84,6OCC_1:84
	]:7.44
]

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 \((0.85)(0.8)(\frac{474 }{\log_{20} 474}-\frac{153}{\log_{20}153})=94.7\)
Status Protein1 Protein2 d d1/2
Query variables 1TOO_1 4JTV_1 124 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]