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Parikh vectors
6NUC_1 1HVA_1 5VCP_1 Letter Amino acid
10 2 9 M Methionine
23 17 14 V Valine
21 13 11 A Alanine
12 11 14 H Histidine
22 17 15 P Proline
18 18 5 S Serine
23 19 16 D Aspartic acid
10 11 5 Q Glutamine
21 24 6 K Lycine
23 12 11 F Phenylalanine
18 12 6 T Threonine
4 7 1 W Tryptophan
20 10 6 N Asparagine
23 13 12 E Glutamic acid
19 22 15 G Glycine
17 9 11 I Isoleucine
41 26 13 L Leucine
18 8 4 Y Tyrosine
18 7 9 R Arginine
11 2 2 C Cysteine 

6NUC_1|Chain A|Serine/threonine-protein phosphatase 2B catalytic subunit alpha isoform|Homo sapiens (9606)
>1HVA_1|Chain A|CARBONIC ANHYDRASE II|Homo sapiens (9606)
>5VCP_1|Chains A, B, C, D|Peptide deformylase|Paraburkholderia xenovorans (strain LB400) (266265)
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 \(w=f(c)\)
6NUC , Knot 162 372 0.86 40 230 364 
GHMSEPKAIDPKLSTTDRVVKAVPFPPSHRLTAKEVFDNDGKPRVDILKAHLMKEGRLEESVALRIITEGASILRQEKNLLDIDAPVTVCGDIHGQFFDLMKLFEVGGSPANTRYLFLGDYVDRGYFSIECVLYLWALKILYPKTLFLLRGNHECRHLTEYFTFKQECKIKYSERVYDACMDAFDCLPLAALMNQQFLCVHGGLSPEINTLDDIRKLDRFKEPPAYGPMCDILWSDPLEDFGNEKTQEHFTHNTVRGCSYFYSYPAVCEFLQHNNLLSILRAHEAQDAGYRMYRKSQTTGFPSLITIFSAPNYLDVYNNKAAVLKYENNVMNIRQFNCSPHPYWLPNFMDVFTWSLPFVGEKVTEMLVNVLN
1HVA , Knot 112 260 0.79 40 177 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFCFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
5VCP , Knot 88 185 0.82 40 142 179 
MAHHHHHHMIREILKMGDPRLLRIADPVDHFDTPELHELVKDMFETMHDANGAGLAAPQIGVNLQVVIFGFGHNERYPDAPPVPETVLINPTITPVSQDMEEGWEGCLSVPGLRGAVSRFSMIKYHGFDQYGKPIDRVAEGFHARVVQHECDHLIGKLYPMRINDFAKFGFTEVLFPDMDPNSDD

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 \(c\).

\(|P_{f(6NUC_1)}(2) \setminus P_{f(1HVA_1)}(2)|=115\), \(|P_{f(1HVA_1)}(2) \setminus P_{f(6NUC_1)}(2)|=62\). 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 1:101001011010100000110111111000101001100010101011010110010100011101100110110000011010111010101010110110110111011000011110010010101001101111011010011110100000010001010000010000010010101100111111100011010111010100100100100100111011100111001100110000000100001010001000111001100001101101001001100100000001110110110110010100001111000001101001000101011101101101011111001001110110
Pair \(Z_2\) Length of longest common subsequence
6NUC_1,1HVA_1 177 4
6NUC_1,5VCP_1 188 3
1HVA_1,5VCP_1 175 3

Newick tree

 
[
	6NUC_1:92.52,
	[
		1HVA_1:87.5,5VCP_1:87.5
	]:5.02
]

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 \((0.85)(0.8)(\frac{632 }{\log_{20} 632}-\frac{260}{\log_{20}260})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 6NUC_1 1HVA_1 135 112.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]