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Parikh vectors
2HQK_1 5WMU_1 6PYK_1 Letter Amino acid
10 28 18 A Alanine
17 26 8 E Glutamic acid
11 19 10 F Phenylalanine
9 29 23 S Serine
12 12 4 Y Tyrosine
16 26 15 V Valine
17 23 8 D Aspartic acid
0 8 2 C Cysteine
13 18 16 I Isoleucine
7 10 1 M Methionine
11 23 9 P Proline
13 15 12 T Threonine
11 16 27 N Asparagine
2 18 14 Q Glutamine
11 53 24 L Leucine
8 17 6 R Arginine
20 29 29 G Glycine
8 17 10 H Histidine
23 33 6 K Lycine
2 5 0 W Tryptophan 

2HQK_1|Chain A|Cyan fluorescent chromoprotein|Clavularia sp. (86521)
>5WMU_1|Chains A, B|Indoleamine 2,3-dioxygenase 1|Homo sapiens (9606)
>6PYK_1|Chain A|Hemolysin|Proteus mirabilis (584)
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 \(w=f(c)\)
2HQK , Knot 103 221 0.83 38 157 216 
GVIKPDMKIKLKMEGNVNGHAFVIEGEGEGKPYDGTNTINLEVKEGAPLPFSYDILTTAFAYGNRAFTKYPDDIPNYFKQSFPEGYSWERTMTFEDKGIVKVKSDISMEEDSFIYEIHLKGENFPPNGPVMQKKTTGWDASTERMYVRDGVLKGDVKHKLLLEGGGHHRVDFKTIYRAKKAVKLPDYHFVDHRIEILNHDKDYNKVTVYESAVARNSTDGM
5WMU , Knot 182 425 0.86 40 233 394 
MSKEYHIDEEVGFALPNPQENLPDFYNDWMFIAKHLPDLIESGQLRERVEKLNMLSIDHLTDHKSQRLARLVLGCITMAYVWGKGHGDVRKVLPRNIAVPYCQLSKKLELPPILVYADCVLANWKKKDPNKPLTYENMDVLFSFRDGDCSKGFFLVSLLVEIAAASAIKVIPTVFKAMQMQERDTLLKALLEIASCLEKALQVFHQIHDHVNPKAFFSVLRIYLSGWKGNPQLSDGLVYEGFWEDPKEFAGGSAGQSSVFQCFDVLLGIQQTAGGGHAAQFLQDMRRYMPPAHRNFLCSLESNPSVREFVLSKGDAGLREAYDACVKALVSLRSYHLQIVTKYILIPASQQPKENKTSEDPSKLEAKGTGGTDLMNFLKTVRSTTEKSLLKEGKGELNSKLEGKPIPNPLLGLDSTRTGHHHHHH
6PYK , Knot 106 242 0.80 38 150 228 
NGIVPDAGHQGPDVSAVNGGTQVINIVTPNNEGISHNQYQDFNVGKPGAVLNNALEAGQSQLAGHLNANSNLNGQAASLILNEVVSRNPSFLLGQQEVFGIAAEYVLSNPNGITCDGCGFINTSRSSLVVGNPLFENGQLKGYSTLNNTNLLSLGKNGLNTTGLLDLIAPRIDSRGKITAAEISAFTGQNTFSQHFDILSSQKPVSALDSYFFGSMQSGRIRIINTAEGSGVKLAGHHHHHH

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 \(c\).

\(|P_{f(2HQK_1)}(2) \setminus P_{f(5WMU_1)}(2)|=49\), \(|P_{f(5WMU_1)}(2) \setminus P_{f(2HQK_1)}(2)|=125\). 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 1:11101010101010101010111101010101001000101010011111100011001110100110001001100100011010010001010001110100010100001100101010011101111000001101000010100111010100011101110001010010010011011000110001011000000001010001110000011
Pair \(Z_2\) Length of longest common subsequence
2HQK_1,5WMU_1 174 4
2HQK_1,6PYK_1 157 3
5WMU_1,6PYK_1 183 7

Newick tree

 
[
	5WMU_1:92.59,
	[
		2HQK_1:78.5,6PYK_1:78.5
	]:14.09
]

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 \((0.85)(0.8)(\frac{646 }{\log_{20} 646}-\frac{221}{\log_{20}221})=119.\)
Status Protein1 Protein2 d d1/2
Query variables 2HQK_1 5WMU_1 155 116
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]