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Parikh vectors
7GKU_1 3QNX_1 4CTE_1 Letter Amino acid
17 11 14 A Alanine
14 14 15 Q Glutamine
9 12 12 E Glutamic acid
27 14 16 V Valine
10 2 6 M Methionine
13 13 7 P Proline
16 30 13 S Serine
3 2 10 W Tryptophan
11 10 7 R Arginine
21 7 19 N Asparagine
17 11 18 D Aspartic acid
12 5 1 C Cysteine
7 3 5 H Histidine
24 17 8 T Threonine
11 8 11 Y Tyrosine
26 15 28 G Glycine
11 7 13 I Isoleucine
29 19 11 L Leucine
11 12 8 K Lycine
17 7 11 F Phenylalanine 

7GKU_1|Chains A, B|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3QNX_1|Chain A|Fab fragment of IMMUNOGLOBULIN A1 LIGHT CHAIN|Homo sapiens (9606)
>4CTE_1|Chains A, B|ENDO-1,3-BETA-GLUCANASE, FAMILY GH16|ZOBELLIA GALACTANIVORANS (63186)
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 \(w=f(c)\)
7GKU , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
3QNX , Knot 104 219 0.85 40 159 210 
DIVMTQSPLSLSVTPGEPASISCRSSQSLLRRDGHNDLEWYLQKPGQSPQPLIYLGSTRASGVPDRFSGSGSGTDFTLKIIRVEAEDAGTYYCMQNKQTPLTFGQGTRLEIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGEC
4CTE , Knot 110 233 0.85 40 167 228 
QDYNLVWQDEFDDGIGPDWVFETGMGYNGWGNNELQYYRRENAAVENGNLVITAKHENFGGAQYTSARMKTQGRKSFKYGKIEARIALPSGQGLWPAFWMLGNNITSVSWPACGEIDIMSRINNALQTHGTIHWSDQNGDHASYGDDVGVSDPGQYHIYSVEWDANSIKWFVDGQQFNEVDISNGVNGTGEFQNEFFILLNMAVGGDWPGFDVDQSKLPAQMLVDYVRVYQKG

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 \(c\).

\(|P_{f(7GKU_1)}(2) \setminus P_{f(3QNX_1)}(2)|=116\), \(|P_{f(3QNX_1)}(2) \setminus P_{f(7GKU_1)}(2)|=79\). 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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair \(Z_2\) Length of longest common subsequence
7GKU_1,3QNX_1 195 3
7GKU_1,4CTE_1 183 3
3QNX_1,4CTE_1 196 4

Newick tree

 
[
	3QNX_1:99.74,
	[
		7GKU_1:91.5,4CTE_1:91.5
	]:8.24
]

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 \((0.85)(0.8)(\frac{525 }{\log_{20} 525}-\frac{219}{\log_{20}219})=87.9\)
Status Protein1 Protein2 d d1/2
Query variables 7GKU_1 3QNX_1 114 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]