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Parikh vectors
5JDE_1 3ONV_1 6SJL_1 Letter Amino acid
28 14 36 K Lycine
9 11 22 M Methionine
11 19 55 D Aspartic acid
6 2 43 Q Glutamine
22 13 51 E Glutamic acid
16 15 75 L Leucine
19 9 66 S Serine
3 0 7 W Tryptophan
6 6 23 Y Tyrosine
4 4 5 C Cysteine
10 23 68 G Glycine
10 12 27 F Phenylalanine
7 6 41 P Proline
4 12 50 R Arginine
9 6 19 H Histidine
29 20 49 V Valine
17 26 64 A Alanine
12 7 29 N Asparagine
17 18 51 I Isoleucine
26 14 60 T Threonine 

5JDE_1|Chains A, B|Titin|Homo sapiens (9606)
>3ONV_1|Chains A, B, C|Purine nucleoside phosphorylase deoD-type|Escherichia coli (83333)
>6SJL_1|Chains A, B, C|Putative type VI secretion protein|Escherichia coli (562)
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 \(w=f(c)\)
5JDE , Knot 115 265 0.80 40 161 245 
GAMVKIIKKPKDVTALENATVAFEVSVSHDTVPVKWFHKSVEIKPSDKHRLVSERKVHKLMLQNISPSDAGEYTAVVGQLECKAKLFVETLHITKTMKNIEVPETKTASFECEVSHFNVPSMWLKNGVEIEMSEKFKIVVQGKLHQLIIMNTSTEDSAEYTFVCGNDQVSATLTVTPIMITSMLKDINAEEKDTITFEVTVNYEGISYKWLKNGVEIKSTDKCQMRTKKLTHSLNIRNVHFGDAADYTFVAGKATSTATLYVVEA
3ONV , Knot 110 237 0.84 38 157 226 
ATPHINAEMGDFADVVLMPGDPLRAKYIAETFLEDAREVNNVRGMLGFTGTYKGRKISVMGHGMGIPSCSIYTKELITDFGVKKIIRVGSCGAVLPHVKLRDVVIGMGACTDSKVNRIRFKDHDFAAIADFDMVRNAVDAAKALGIDARVGNLFSADLFYSPDGEMFDVMEKYGILGVEMEAAGIYGVAAEFGAKALTICTVSDHIRTHEQTTAAERQTTFNDMIKIALESVLLGDK
6SJL , Knot 313 841 0.83 40 300 757 
MNLTDSLQNVLSGVGTLNRYRLDIPSCPSSLDVEDFSGTEGISKIYRYDIIFTSTDRYPDAAWFLRKSATLIMGTGLLESLTEQKKVHGVITDFRRISGSEDQAQYRITLKPFISLLDKQFRTHRFFVNKSVPEVVEQILTEHGLKGWEYEFSLKRTYPKREQINQYQESDLRFIQRLLAEVGIFYFFTLHPDAQTEVIHFGDVQAALIFDKTLPVNSPSGMSDSGTDSVWALNVEHRVVESRVITNDYNHREAQNILMSVPADMTRGEGYDTSYGEVYHYRPRHTERGDKIDPAPETANFWARLDHERFLAEQTRITGKSTDASLLPAQVLTITDSTPPVLPAPLQEPVLLTQLLFTGSRKSALQVMLSAVPYSEIVCWRPPLLTRPKITGTMTARVTSAKEGDIYAWQDASGMYRVKFDADRDDKNPGQESMPVRLAKPYSGDAYGFHFPLIQGTEVAIAFEEGDPDRPYIAHALHDSRHVDHVTDKNGTRNVIRTPANNKLRMEDKRGEEHIKLSTEYGGKTQLNLGHNVDASRELRGEGAELRTDDWISIRGGKGIFISADMQPQAQGKMLDMDEAIRQLEQALSLARSMAKAATAANATQGDISCQQRLNASLTDLTAPGMLLHAPDGIGMVSARALRIASGSESVGIMSGDNTDITAGQSFTVVAEGAVSLLSRNQGMQLLAAKGRVNIQAQSDDLSMSSQQNLDIQSSEGKVTVSANQELILACGGAYIKLSGGNIELGCPGQILLKSTNMQKMGPTSLDIASVEMPRGFGGGFILTDEAGVPQPSTPYRLTTAEGDILQGITDENGKTAPVNTSIPSVVKVEFGKVKIHGETE

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 \(c\).

\(|P_{f(5JDE_1)}(2) \setminus P_{f(3ONV_1)}(2)|=82\), \(|P_{f(3ONV_1)}(2) \setminus P_{f(5JDE_1)}(2)|=78\). 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 1:1111011001001011001011101010000111011000101010000011000010011100101001100011110100010111001010001001011000010100010010110111001101010001011101010011110000000100011010001010101011110011001010000010101010001100011001101000000010000100010100101101100011110100010101101
Pair \(Z_2\) Length of longest common subsequence
5JDE_1,3ONV_1 160 4
5JDE_1,6SJL_1 199 4
3ONV_1,6SJL_1 175 4

Newick tree

 
[
	6SJL_1:97.83,
	[
		5JDE_1:80,3ONV_1:80
	]:17.83
]

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 \((0.85)(0.8)(\frac{502 }{\log_{20} 502}-\frac{237}{\log_{20}237})=76.1\)
Status Protein1 Protein2 d d1/2
Query variables 5JDE_1 3ONV_1 94 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]