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Parikh vectors
7TIN_1 2CHR_1 8WEN_1 Letter Amino acid
4 3 4 W Tryptophan
23 9 21 Y Tyrosine
29 24 22 E Glutamic acid
27 29 23 G Glycine
31 19 24 T Threonine
36 21 30 D Aspartic acid
4 3 5 C Cysteine
59 39 34 L Leucine
13 10 12 M Methionine
25 14 12 P Proline
39 41 33 A Alanine
21 22 15 R Arginine
34 9 25 N Asparagine
27 36 22 V Valine
48 22 25 I Isoleucine
28 7 15 F Phenylalanine
39 31 31 S Serine
22 13 11 Q Glutamine
15 6 8 H Histidine
35 12 25 K Lycine 

7TIN_1|Chains A, B, C, D|2-succinyl-5-enolpyruvyl-6-hydroxy-3-cyclohexene-1-carboxylate synthase|Staphylococcus aureus (1280)
>2CHR_1|Chain A|CHLOROMUCONATE CYCLOISOMERASE|Cupriavidus necator (106590)
>8WEN_1|Chains A, B[auth C]|Metalloprotease|Bacteroides fragilis (817)
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 \(w=f(c)\)
7TIN , Knot 224 559 0.84 40 256 518 
GAMGNHKAALTKQVFTFASELYAYGVREVVISPGSRSTPLALAFEAHPNIKTWIHPDERSAAFFAVGLIKGSERPVAILCTSGTAAANYTPAIAESQISRIPLIVLTSDRPHELRSVGAPQAINQVNMFNNYVSYEFDMPIADDSKETINAIYYQMQIASQYLYGPHKGPIHFNLPFRDPLTPDLNATELLTSEMKILPHYQKSIDASALRHILNKKKGLIIVGDMQHQEVDQILTYSTIYDLPILADPLSHLRKFDHPNVICTYDLLFRSGLDLNVDFVIRVGKPVISKKLNQWLKKTDAFQILVQNNDKIDVFPIAPDISYEISANDFFRSLMEDTTINRVSWLEKWQCLEKKGRKEIKCYLEQATDESAFVGELIKKTSEKDALFISNSMPIRDVDNLLLNKNIDVYANRGANGIDGIVSTALGMAVHKRITLLIGDLSFYHDMNGLLMSKLNNIQMNIVLLNNDGGGIFSYLPQKESATDYFERLFGTPTGLDFEYTAKLYQFDFKRFNSVSEFKNATLLSETSTIYELITNREDNFKQHQILYQKLSEMIHDTL
2CHR , Knot 157 370 0.83 40 199 345 
MKIDAIEAVIVDVPTKRPIQMSITTVHQQSYVIVRVYSEGLVGVGEGGSVGGPVWSAECAETIKIIVERYLAPHLLGTDAFNVSGALQTMARAVTGNASAKAAVEMALLDLKARALGVSIAELLGGPLRSAIPIAWTLASGDTKRDLDSAVEMIERRRHNRFKVKLGFRSPQDDLIHMEALSNSLGSKAYLRVDVNQAWDEQVASVYIPELEALGVELIEQPVGRENTQALRRLSDNNRVAIMADESLSTLASAFDLARDRSVDVFSLKLCNMGGVSATQKIAAVAEASGIASYGGTMLDSTIGTSVALQLYSTVPSLPFGCELIGPFVLADTLSHEPLEIRDYELQVPTGVGHGMTLDEDKVRQYARVS
8WEN , Knot 170 397 0.85 40 231 380 
MKNVKLLLMLGTAALLAACSNEADSLTTSIDTPVTASIDLQSVSYTDLATQLNDVSDFGKMIILKDNGFNRQVHVSMDKRTKIQLDNENVRLFNGRDKDSTSFILGDEFAVLRFYRNGESISYIAYKEAQMMNEIAEFYAAPFKKTRAINEKEAFECIYDSRTRSAGKDLVSVKINIDKAKKILNLPECDYINDYIKTPQVPHGITESQTRAVPSEPKTVYVICLRESGSTVYPNEVSAQMQDAANSVYAVHGLKRFVNLHFVLYTTEYSCPSGNADEGLDGFTASLKANPKAEGYDDQIYFLIRWGTWDNNILGISWLDSYNVNTASDFKASGMSTTQLMYPGVMAHELGHILGARHADDPKDLMYSKYTGYLFHLSEENMYRIAKNLGWEIADGD

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 \(c\).

\(|P_{f(7TIN_1)}(2) \setminus P_{f(2CHR_1)}(2)|=103\), \(|P_{f(2CHR_1)}(2) \setminus P_{f(7TIN_1)}(2)|=46\). 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 1:1111000111000110110010101100111011000011111101010100110100001111111110100011111000101110001111000100111111000010010011110110010110001000101111000000101100010110001011001110101110011010101001100010111000001010110011000011111101000010011000010011111011001001001011000011100110101011101101110001001100001101110000010111111010001010011001100001001011001001000100010001001000011110110000000111100011100100111000101010011011011100111111000101111010100010111100100101011110001111100110000100010011101011010001010010100100100100101100000100110000001000011000100110001
Pair \(Z_2\) Length of longest common subsequence
7TIN_1,2CHR_1 149 4
7TIN_1,8WEN_1 147 4
2CHR_1,8WEN_1 150 3

Newick tree

 
[
	2CHR_1:75.16,
	[
		7TIN_1:73.5,8WEN_1:73.5
	]:1.66
]

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 \((0.85)(0.8)(\frac{929 }{\log_{20} 929}-\frac{370}{\log_{20}370})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 7TIN_1 2CHR_1 187 154.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]