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Parikh vectors
7BAL_1 8RMQ_1 3MWO_1 Letter Amino acid
12 17 1 C Cysteine
26 35 22 G Glycine
17 36 12 F Phenylalanine
16 49 18 S Serine
21 32 10 N Asparagine
14 18 11 Q Glutamine
11 52 24 K Lycine
3 12 7 W Tryptophan
11 19 8 Y Tyrosine
17 36 13 A Alanine
17 37 19 D Aspartic acid
9 78 13 E Glutamic acid
11 49 9 I Isoleucine
29 63 26 L Leucine
24 39 12 T Threonine
27 31 17 V Valine
11 43 7 R Arginine
10 24 2 M Methionine
13 33 17 P Proline
7 13 12 H Histidine 

7BAL_1|Chain A|Main Protease|Severe acute respiratory syndrome coronavirus 2 (2697049)
>8RMQ_1|Chain A|Polymerase acidic protein|Influenza A virus (A/Zhejiang/DTID-ZJU01/2013(H7N9)) (1318616)
>3MWO_1|Chains A, B|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 \(w=f(c)\)
7BAL , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
8RMQ , Knot 280 716 0.85 40 298 662 
MEDFVRQCFNPMIVELAEKAMKEYGEDPKIETNKFASICTHLEVCFMYSDFHFIDERGESTIIESGDPNVLLKHRFEIIEGRDRTMAWTVVNSICNTTGVEKPKFLPDLYDYKENRFIEIGVTRREVHIYYLEKANKIKSEKTHIHIFSFTGEEMATKADYTLDEESRARIKTRLFTIRQEMASRGLWDSFRQSERGEETIEERFEITGTMRRLADQSLPPNFSSLENFRAYVDGFEPNGCIEGKLSQMSKEVNARIEPFLRTTPRPLRLPDGPPCSQRSKFLLMDALKLSIEDPSHEGEGIPLYDAIKCMKTFFGWKEPNIIKPHEKGINPNYLLTWKQVLAELQDIKNEEKIPRTKNMKKTSQLKWALGENMAPEKVDFEDCKDVNDLKQYDSDEPEPRSLACWIQSEFNKACELTDSSWVELDEIGEDVAPIEHIASMRRNYFTAEVSHCRATEYIMKGVYINTALLNASCAAMDDFQLIPMISKCITKEGRRKTNLYGFIIKGRSHLRNDTDVVNFVSMEFSLTDPRLEPHKWEKYCVLEIGDMLLRTAVGQVSRPMFLYVRTNGTSKIKMKWGMEMRRCLLQSLQQIESMIEAESSVKEKDLTKEFFENKSETWPIGESPKGVEEGSIGKVCRTLLAKSVFNSLYASPQLEGFSAESRKLLLIVQALRDNLEPGTFDLEGLYEAIEECLINDPWVLLNASWFNSFLTHALR
3MWO , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(7BAL_1)}(2) \setminus P_{f(8RMQ_1)}(2)|=41\), \(|P_{f(8RMQ_1)}(2) \setminus P_{f(7BAL_1)}(2)|=143\). 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
7BAL_1,8RMQ_1 184 4
7BAL_1,3MWO_1 170 3
8RMQ_1,3MWO_1 188 3

Newick tree

 
[
	8RMQ_1:95.52,
	[
		7BAL_1:85,3MWO_1:85
	]:10.52
]

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 \((0.85)(0.8)(\frac{1022 }{\log_{20} 1022}-\frac{306}{\log_{20}306})=191.\)
Status Protein1 Protein2 d d1/2
Query variables 7BAL_1 8RMQ_1 243 172
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]