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Parikh vectors
6NLI_1 1IFU_1 5WLT_1 Letter Amino acid
10 4 11 H Histidine
5 4 1 M Methionine
2 12 12 F Phenylalanine
5 24 13 A Alanine
9 13 11 Q Glutamine
3 17 12 T Threonine
5 14 8 Y Tyrosine
5 16 17 V Valine
5 21 7 R Arginine
27 22 26 L Leucine
12 2 24 K Lycine
1 15 17 P Proline
5 17 17 S Serine
2 2 1 C Cysteine
18 15 13 E Glutamic acid
9 17 22 G Glycine
12 22 9 I Isoleucine
2 1 7 W Tryptophan
8 17 10 N Asparagine
13 8 19 D Aspartic acid 

6NLI_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Ferroxidase|Pseudomonas aeruginosa (strain ATCC 15692 / PAO1 / 1C / PRS 101 / LMG 12228) (208964)
>1IFU_1|Chain A|RICIN|Ricinus communis (3988)
>5WLT_1|Chain A|Carbonic anhydrase 2|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 \(w=f(c)\)
6NLI , Knot 80 158 0.85 40 111 156 
MKGDKKVIQHLNKILGNELIAINQYFLHSRMWNDWGLKRLGAHEYHESIDEMKHADKLIERILFLEGLPNLQDLGKLLIGENTQEMLQCDLNLELKATKDLREAIVHCEQVHDYVSRDLLKDILESEEEHIDYLETQLGLIQKVGLENYLQSHMHEDD
1IFU , Knot 116 263 0.82 40 167 255 
MVPKQYPIINFTTAGATVQSYTNFIRAVRGRLTTGADVRHEIPVLPNRVGLPINQRFILVELSNHAELSVTLALDVTNAYVVGYRAGNSAYFFHPDNQEDAEAITHLFTDVQNRYTFAFGGNYDRLEQLAGNLRENIELGNGPLEEAISALYYYSTGGTQLPTLARSFIICIQMISEAARFQYIEGEMRTRIRYNRRSAPDPSVITLENSWGRLSTAIQESNQGAFASPIQLQRRNGSKFSVYDVSILIPIIALMVYRCAPPP
5WLT , Knot 110 257 0.79 40 173 246 
HWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(6NLI_1)}(2) \setminus P_{f(1IFU_1)}(2)|=60\), \(|P_{f(1IFU_1)}(2) \setminus P_{f(6NLI_1)}(2)|=116\). 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 1:10100011001001110011110001100011001110011100000010010010011001111011101001101111000001100010101010001001110000100010001100110000001001000111100111000100010000
Pair \(Z_2\) Length of longest common subsequence
6NLI_1,1IFU_1 176 3
6NLI_1,5WLT_1 170 3
1IFU_1,5WLT_1 176 3

Newick tree

 
[
	1IFU_1:88.97,
	[
		6NLI_1:85,5WLT_1:85
	]:3.97
]

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 \((0.85)(0.8)(\frac{421 }{\log_{20} 421}-\frac{158}{\log_{20}158})=78.3\)
Status Protein1 Protein2 d d1/2
Query variables 6NLI_1 1IFU_1 101 80.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]