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Parikh vectors
6RGI_1 1ZIO_1 6QXL_1 Letter Amino acid
8 18 60 A Alanine
7 8 9 N Asparagine
2 5 5 C Cysteine
12 22 37 G Glycine
4 6 11 F Phenylalanine
3 16 35 R Arginine
4 17 38 D Aspartic acid
4 2 26 S Serine
1 0 2 W Tryptophan
2 10 12 Q Glutamine
7 14 28 E Glutamic acid
4 12 27 I Isoleucine
8 22 43 L Leucine
4 4 12 H Histidine
16 12 26 K Lycine
2 10 15 M Methionine
4 9 19 P Proline
8 9 24 T Threonine
5 7 9 Y Tyrosine
3 14 45 V Valine 

6RGI_1|Chain A|Cytochrome c iso-1|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>1ZIO_1|Chain A|ADENYLATE KINASE|Geobacillus stearothermophilus (1422)
>6QXL_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Pyruvate kinase|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
6RGI , Knot 58 108 0.83 40 92 105 
AEFKAGSAKKGATLFKTRCLQCHTVEKGGPHKVGPNLHGIFGRHSGQAEGYSYTDANIKKNVLWDENNMSEYLTNPKKYIPGTKMAFGGLKKEKDRNDLITYLKKATE
1ZIO , Knot 100 217 0.82 38 155 210 
MNLVLMGLPGAGKGTQAEKIVAAYGIPHISTGDMFRAAMKEGTPLGLQAKQYMDRGDLVPDEVTIGIVRERLSKDDCQNGFLLDGFPRTVAQAEALETMLADIGRKLDYVIHIDVRQDVLMERLTGRRICRNCGATYHLIFHPPAKPGVCDKCGGELYQRADDNEATVANRLEVNMKQMKPLVDFYEQKGYLRNINGEQDMEKVFADIRELLGGLAR
6QXL , Knot 193 483 0.82 40 224 449 
MSVRRTKIVATLGPASNSPEVLEQLILAGIDVARLNFSHGTPDEHRARARLVRELAAKHGRFVALLGDLQGPKIRIAKFANKRIELQVGDKFRFSTSHARDAGTQEVVGIDYPDLVKDCGVGDELLLDDGRVVMVVEEVAADELRCRVLIGGPLSDHKGINRRGGGLTAPALTDKDKADIKLAADMDLDYVAVSFPRDAKDMEYARRLLTEAGGKAWLVAKIERAEAVADDDALDGLIRASDAVMVARGDLGVEIGDAELVGIQKKIILHARRNNKVVITATQMMESMIHSPMPTRAEVSDVANAVLDYTDAVMLSAESAAGEYPVEAVKAMARVCQGAEKHPTSQKSSHRLGQTFDRCDESIALASMYTANHFPGIKAIICLTESGFTPLIMSRIRSSVPIYAYSPHRETQARVAMFRGVETIPFDPAALPAEKVSQAAVDELLKRGVVTKGDWVILTKGDSYTAQGGTNTMKVLHVGDLLV

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 \(c\).

\(|P_{f(6RGI_1)}(2) \setminus P_{f(1ZIO_1)}(2)|=55\), \(|P_{f(1ZIO_1)}(2) \setminus P_{f(6RGI_1)}(2)|=118\). 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 1:101011010011011000010000100111001110101111000101010000010100011100001000100100011100111111000000001100100100
Pair \(Z_2\) Length of longest common subsequence
6RGI_1,1ZIO_1 173 3
6RGI_1,6QXL_1 198 4
1ZIO_1,6QXL_1 157 4

Newick tree

 
[
	6RGI_1:97.30,
	[
		1ZIO_1:78.5,6QXL_1:78.5
	]:18.80
]

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 \((0.85)(0.8)(\frac{325 }{\log_{20} 325}-\frac{108}{\log_{20}108})=67.4\)
Status Protein1 Protein2 d d1/2
Query variables 6RGI_1 1ZIO_1 86 64
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]