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Parikh vectors
4JJZ_1 4HYG_1 7RNM_1 Letter Amino acid
33 6 13 D Aspartic acid
10 4 9 Q Glutamine
9 2 13 H Histidine
36 5 11 K Lycine
15 13 15 M Methionine
34 12 11 T Threonine
17 8 4 Y Tyrosine
25 8 11 R Arginine
7 0 4 C Cysteine
25 18 9 P Proline
21 16 19 S Serine
64 36 15 A Alanine
17 18 7 F Phenylalanine
2 4 3 W Tryptophan
23 6 8 N Asparagine
32 9 15 E Glutamic acid
54 29 10 G Glycine
37 22 12 I Isoleucine
55 53 48 L Leucine
43 32 13 V Valine 

4JJZ_1|Chains A, B|Formate--tetrahydrofolate ligase|Moorella thermoacetica (264732)
>4HYG_1|Chains A, B, C, D|Putative uncharacterized protein|Methanoculleus marisnigri JR1 (368407)
>7RNM_1|Chains A, B|Estrogen receptor|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 \(w=f(c)\)
4JJZ , Knot 219 559 0.82 40 237 511 
MSKVPSDIEIAQAAKMKPVMELARGLGIQEDEVELYGKYKAKISLDVYRRLKDKPDGKLILVTAITPTPAGEGKTTTSVGLTDALARLGKRVMVCLREPSLGPSFGIKGGAAGGGYAQVVPMEDINLHFTGDIHAVTYAHNLLAAMVDNHLQQGNVLNIDPRTITWRRVIDLNDRALRNIVIGLGGKANGVPRETGFDISVASEVMACLCLASDLMDLKERFSRIVVGYTYDGKPVTAGDLEAQGSMALLMKDAIKPNLVQTLENTPAFIHGGPFANIAHGCNSIIATKTALKLADYVVTEAGFGADLGAEKFYDVKCRYAGFKPDATVIVATVRALKMHGGVPKSDLATENLEALREGFANLEKHIENIGKFGVPAVVAINAFPTDTEAELNLLYELCAKAGAEVALSEVWAKGGEGGLELARKVLQTLESRPSNFHVLYNLDLSIKDKIAKIATEIYGADGVNYTAEADKAIQRYESLGYGNLPVVMAKTQYSFSDDMTKLGRPRNFTITVREVRLSAGAGFIVPITGAIMTMPGLPKRPAACNIDIDADGVITGLF
4HYG , Knot 122 301 0.77 38 152 275 
MQIRDWLPLLGMPLMLLFVQIIAIVLVMPMQAAGLVAFENPSSVANPLIFIGMLLAFTLVLLVLLRTGGRRFIAAFIGFALFMTFLYIFGALSLLALGPTTAAAAGTLIGAVAVTALLYLYPEWYVIDILGVLISAGVASIFGISLEPLPVLVLLVLLAVYDAISVYRTKHMITLAEGVLETKAPIMVVVPKRADYSFRKEGLNISEGEERGAFVMGMGDLIMPSILVVSSHVFVDAPAVLWTLSAPTLGAMVGSLVGLAVLLYFVNKGNPQAGLPPLNGGAILGFLVGAALAGSFSWLPF
7RNM , Knot 114 250 0.84 40 162 239 
SLALSLTADQMVSALLDAEPPILYSEYDPTRPFSEASMMGLLTNLADRELVHMINWAKRVPGFVDLTLHDQVHLLECAWLEILMIGLVWRSMEHPGKLLFAPNLLLDRNQGKCVEGMVEIFDMLLATSSRFRMMNLQGEEFVCLKSIILLNSGVYTFLSSTLKSLEEKDHIHRVLDKITDTLIHLMAKAGLTLQQQHQRLAQLLLILSHIRHMSNKGMEHLYSMKCKNVVPLSDLLLEMLDAHRLHAPTS

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 \(c\).

\(|P_{f(4JJZ_1)}(2) \setminus P_{f(4HYG_1)}(2)|=125\), \(|P_{f(4HYG_1)}(2) \setminus P_{f(4JJZ_1)}(2)|=40\). 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 1:1001100101101101011101101111000010101000101010100010001010111101101011101000001110011101100111010010111011101111111010111100101010101011001001111110001001011010100101001101000110011111110101110001101011001110101100110100010011110000101101101010101111100110101100100011110111110110100011100011011001100111110111001001000011101010111101011010111100011000101100111010001001101111111110111000010101100101011101110011101101110110011001000100101100101010001101100101101100010100110000011010111111000001000100110100101010010101111111110111101111100111001010101110111
Pair \(Z_2\) Length of longest common subsequence
4JJZ_1,4HYG_1 165 3
4JJZ_1,7RNM_1 175 4
4HYG_1,7RNM_1 174 4

Newick tree

 
[
	7RNM_1:88.77,
	[
		4JJZ_1:82.5,4HYG_1:82.5
	]:6.27
]

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 \((0.85)(0.8)(\frac{860 }{\log_{20} 860}-\frac{301}{\log_{20}301})=151.\)
Status Protein1 Protein2 d d1/2
Query variables 4JJZ_1 4HYG_1 192 144.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]