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Parikh vectors
9JJF_1 7LMJ_1 1KBO_1 Letter Amino acid
99 26 21 G Glycine
93 12 19 I Isoleucine
148 30 25 L Leucine
96 16 17 S Serine
121 24 14 V Valine
87 12 9 R Arginine
105 9 17 E Glutamic acid
115 17 25 A Alanine
67 19 9 N Asparagine
88 11 24 K Lycine
53 10 6 M Methionine
68 13 13 P Proline
70 26 9 T Threonine
14 12 1 C Cysteine
70 14 12 Q Glutamine
68 16 17 F Phenylalanine
27 3 6 W Tryptophan
51 11 10 Y Tyrosine
88 17 12 D Aspartic acid
32 8 7 H Histidine 

9JJF_1|Chain A|Transient receptor potential cation channel subfamily M member-like 2|Nematostella vectensis (45351)
>7LMJ_1|Chain A[auth AAA]|3C-like proteinase|Severe acute respiratory syndrome coronavirus (694009)
>1KBO_1|Chains A, B, C, D|NAD(P)H dehydrogenase [quinone] 1|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 \(w=f(c)\)
9JJF , Knot 552 1560 0.86 40 357 1350 
MGKDSFTPLYDGGDSSHVHLNKFGSNQLSQSKKSWIARNFSRRECIRFVPKSHDVSRCKCGRPRERHSQQALESGQGSEEWNVASCTTKHPTNAYGEIDFEGYGGQKRAPYLRMSHDTDANLVITLMLKRWNLEIPNLVISVTGGAKSFVLKPRLREMFRRGLIKAAKTTGAWIITGGTNTGVMKHVGEAVKEQQLMFGSDTQVNVIGIATWGIVDKQSDLISEKNGKYPALYSMEPTPGHQGAMLDPNHSHFFLVDDGTEGKYGVEIGMRSRIEEAIMKVKTDSRSEAGSIGVPVVLLVLEGGPNTVATMYELIKKKVPAVVIDGSGRAASVVGFAYNHTIKRNVDGQTINVIDPQYEDEVRAKVVEVFGAKGADKTYSMIKDVLEDEKMISVYSLDGEISQDIDLAILKALLKANRSSPVAQLNLALAWNRIDLAKSDIFTEEQQWTTETLSAAMLTALLDDKAEFAELFLQNGLSMREFLSLDILCKLYAEVPGNTTIKPLLQKEMGKRQVKTIDMDVVGEVIEELMGDMFESYYRKDGHYFGELASYAEGLVLKNRKSSKDLLANINRIDPLPTPYLDVFLWAVLCNRRELARVLWEAGREPMAAALMASRLLKRMASRAQEDNTITDISSDLYDHARLFEERAVGVLDECFNENETLSQTLLVRELDHYSRMTALELAVSAESQDFIAHTSCQVLLTRLWMGTMAMNTRWWKVLVCLYLPVLIFPIIYFVPDEQHERQAAEREHQKSLNQKSSKVKSHKEKNDAPVVPVYRSKEEKAVSNDEEARVGTENEEEDFQLEDYIPEIREDDSMEVIMRNKKLGFCDRIMHFYSAPFSKFVGNVVGYLAFIFLYAYVVLFNFPRFDPAKTLGGIHPTEIVLYFWVFTILIEEIRQLAAKPPKYIKDKVSVYFSDTWNFVDIFSLTVFIIAIILRFFTNSRIFTASRIILSLDIIFFIVRSLQIFSVNRLLGPKLVMIQKMMQDLAQFIIILAVFTIAYGIALHAVMFPSPGIYARNNTWVTITSVVQYPYWQMYGELFLDEIQGEKPKEFGEVDPDGRWLSPLLLAIYMVFTNILLLNLLIAIFNYTFERVQEDSDKVWKFQRYDLVQEYHSRPVFAPPLVLLGHILIFIRWVWRMCRCGHPPRGSTMKIGLSPAEMEQMDNWEFQAAEMYIHQQQQKNSGTLEERVRALGDRVDCINSQLNRVLDSMSGTRAHALTDGNGLEGGHDSEGRLARMEVELSSNSESLQKILALLQQQPPVKGQAAVPIQLTLLHYKARSSPYPGSTAKRFAVQDNMVDWQVPFPDYKPVNYTAPVVLANPVWADKDLMAMSPRPELPYNQMDHTCNVNRVSYNGTYVVKDGLPLNPMGRTGMQGRGLLGRFGPNHAADPVVTRWKRTSAGVMLQGGKKVLEFVAIQRKDNNQWAIPGGMVEPGQLVTQALKAEFGEEAMAKLNVSQEEKERIAKQIERLFQQGQEIYKGYVDDPRNTDNAWMETVAVNFHDDKGDLFGDITLQAGDDAAAVRWQRVSGNIPLYASHVSILEKVAKMRDAAFSNSLEVLFQ
7LMJ , Knot 133 306 0.83 40 193 288 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDTVYCPRHVICTAEDMLNPNYEDLLIRKSNHSFLVQAGNVQLRVIGHSMQNCLLRLKVDTSNPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNHTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGKFYGPFVDRQTAQAAGTDTTITLNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCAALKELLQNGMNGRTILGSTILEDEFTPFDVVRQCSGVTFQ
1KBO , Knot 122 273 0.83 40 177 264 
VGRRALIVLAHSERTSFNYAMKEAAAAALKKKGWEVVESDLYAMNFNPIISRKDITGKLKDPANFQYPAESVLAYKEGHLSPDIVAEQKKLEAADLVIFQFPLQWFGVPAILKGWFERVFIGEFAYTYAAMYDKGPFRSKKAVLSITTGGSGSMYSLQGIHGDMNVILWPIQSGILHFCGFQVLEPQLTYSIGHTPADARIQILEGWKKRLENIWDETPLYFAPSSLFDLNFQAGFLMKKEVQDEEKNKKFGLSVGHHLGKSIPTDNQIKARK

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 \(c\).

\(|P_{f(9JJF_1)}(2) \setminus P_{f(7LMJ_1)}(2)|=180\), \(|P_{f(7LMJ_1)}(2) \setminus P_{f(9JJF_1)}(2)|=16\). 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 1:110001011001100001010011000100000011100100000101110000100000101000000011001010001011000000100101010101011000110101000001011101110010101101110101110011101010011001110110001111101100011100110110000111100001011111011110000011000010011100101011001111010000111100100100110111000100111010000000110111111111101110011010011000111111010101101111100001000101001011010000010101101111011000001100110000110100101010001011110111010000111010111110010110001100000100001011110111000101101110011010011010110010101110001011100011000100101011101100111011000000010011011001011110000000011101001011101010111111100000110111011001111111100110011001000001001000100010110001111100010000010001110010000010110111010000111000001110011110111000110111010111111111011100000001100000001000000100000000111111000000011000001011000000010100011010000010111000011100011010011100111011101111110101111011010110011110100111011110111001001110110010001010100010110110101111111101100001101001110101111110010110100111101111001100110111111110110111101111101110100001101001100101010101110010100100110101010110111111011100111101111110001001000000110100001100000011111111111011111011101000101101001011101101001001010110101000000001010001011100100100010011001010010110010110110000101101010100000010011111000111010111110101100010001011001001110001101011110001100011111101111000111101010110001000001001000100110011110111001101011110111001101110010000111110110011011110000000111111110110110011010110011101010000000110010011001001001010010000011100111010000101110101011001111010010101110100101100110100111000101110
Pair \(Z_2\) Length of longest common subsequence
9JJF_1,7LMJ_1 196 5
9JJF_1,1KBO_1 194 4
7LMJ_1,1KBO_1 180 4

Newick tree

 
[
	9JJF_1:99.87,
	[
		1KBO_1:90,7LMJ_1:90
	]:9.87
]

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 \((0.85)(0.8)(\frac{1866 }{\log_{20} 1866}-\frac{306}{\log_{20}306})=395.\)
Status Protein1 Protein2 d d1/2
Query variables 9JJF_1 7LMJ_1 508 302
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]