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Parikh vectors
5AUP_1 7BRD_1 7NZS_1 Letter Amino acid
3 5 18 S Serine
15 22 12 A Alanine
12 14 25 K Lycine
2 3 2 M Methionine
8 8 11 F Phenylalanine
4 8 11 H Histidine
7 22 26 L Leucine
3 4 10 Y Tyrosine
8 12 19 D Aspartic acid
3 2 12 Q Glutamine
4 1 1 C Cysteine
19 9 13 E Glutamic acid
10 20 22 G Glycine
9 10 8 I Isoleucine
2 10 17 P Proline
2 9 14 T Threonine
7 11 7 R Arginine
2 11 8 N Asparagine
2 1 7 W Tryptophan
17 12 17 V Valine 

5AUP_1|Chains A, C[auth H]|Probable hydrogenase nickel incorporation protein HypA|Thermococcus kodakaraensis (strain ATCC BAA-918 / JCM 12380 / KOD1) (69014)
>7BRD_1|Chains A, B|Peptidyl-tRNA hydrolase|Klebsiella pneumoniae (573)
>7NZS_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 \(w=f(c)\)
5AUP , Knot 70 139 0.82 40 106 134 
MHEWALADAIVRTVLDYAQREGASRVKAVRVVLGELQDVAEDIVKFAMEQLFAGTIAEGAEIEFVEEEAVFKCRNCNYEWKLKEVKDKFDERIKEDIHFIPEVVHAFLACPKCGSHDFEVVKGRGVYVAGIKIEKEGGS
7BRD , Knot 89 194 0.80 40 136 186 
MTIKLIVGLANPGAEYAATRHNAGAWYVDLLADRHRAPLREESKFFGYTSRINLAGEDVRLLVPTTFMNLSGKAVAAMATFYRINPDEILVAHDELDLPPGVAKFKLGGGHGGHNGLKDIISKLGNNPNFHRLRVGIGHPGDKNKVVGFVLGKPPASEQKLIDDAVDEAARCTEILLKDGLTKATNRLHAFKAQ
7NZS , Knot 111 260 0.79 40 173 248 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNTGYLFQVEFDDSQDKAVLKGGPLDGTYRLKQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDYGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLTECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(5AUP_1)}(2) \setminus P_{f(7BRD_1)}(2)|=57\), \(|P_{f(7BRD_1)}(2) \setminus P_{f(5AUP_1)}(2)|=87\). 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 1:1001111011100110010001100101101111010011001101110011110110110101100011100000000101001000100010001011101101111010010001011010110111101000110
Pair \(Z_2\) Length of longest common subsequence
5AUP_1,7BRD_1 144 3
5AUP_1,7NZS_1 177 3
7BRD_1,7NZS_1 163 4

Newick tree

 
[
	7NZS_1:89.00,
	[
		5AUP_1:72,7BRD_1:72
	]:17.00
]

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 \((0.85)(0.8)(\frac{333 }{\log_{20} 333}-\frac{139}{\log_{20}139})=59.4\)
Status Protein1 Protein2 d d1/2
Query variables 5AUP_1 7BRD_1 75 65
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]