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Parikh vectors
3HQN_1 9JEU_1 1FBR_1 Letter Amino acid
24 7 8 R Arginine
24 3 5 N Asparagine
28 14 6 E Glutamic acid
0 1 4 W Tryptophan
31 4 6 S Serine
27 6 10 T Threonine
25 2 3 Q Glutamine
12 4 2 H Histidine
29 10 3 L Leucine
16 7 1 P Proline
14 7 1 F Phenylalanine
52 11 3 V Valine
46 2 3 A Alanine
29 6 6 D Aspartic acid
14 1 8 C Cysteine
36 8 3 I Isoleucine
36 12 11 G Glycine
27 9 5 K Lycine
13 2 2 M Methionine
16 2 3 Y Tyrosine 

3HQN_1|Chains A, B[auth D]|Pyruvate kinase|Leishmania mexicana (5665)
>9JEU_1|Chains A, B|Cupin type-2 domain-containing protein|Thermotoga maritima (2336)
>1FBR_1|Chain A|FIBRONECTIN|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 \(w=f(c)\)
3HQN , Knot 210 499 0.87 38 255 475 
MSQLAHNLTLSIFDPVANYRAARIICTIGPSTQSVEALKGLIQSGMSVARMNFSHGSHEYHQTTINNVRQAAAELGVNIAIALDTKGPEIRTGQFVGGDAVMERGATCYVTTDPAFADKGTKDKFYIDYQNLSKVVRPGNYIYIDDGILILQVQSHEDEQTLECTVTNSHTISDRRGVNLPGCDVDLPAVSAKDRVDLQFGVEQGVDMIFASFIRSAEQVGDVRKALGPKGRDIMIICKIENHQGVQNIDSIIEESDGIMVARGDLGVEIPAEKVVVAQKILISKCNVAGKPVICATQMLESMTYNPRPTRAEVSDVANAVFNGADCVMLSGETAKGKYPNEVVQYMARICLEAQSALNEYVFFNSIKKLQHIPMSADEAVCSSAVNSVYETKAKAMVVLSNTGRSARLVAKYRPNCPIVCVTTRLQTCRQLNITQGVESVFFDADKLGHDEGKEHRVAAGVEFAKSKGYVQTGDYCVVIHADHKVKGYANQTRILLVE
9JEU , Knot 61 118 0.82 40 89 112 
GPSGMILKRAYDVTPQKISTDKVRGVRKRVLIGLKDAPNFVMRLFTVEPGGLIDRHSHPWEHEIFVLKGKLTVLKEQGEETVEEGFYIFVEPNEIHGFRNDTDSEVEFLCLIPKEGGE
1FBR , Knot 51 93 0.82 40 74 90 
AEKCFDHAAGTSYVVGETWEKPYQGWMMVDCTCLGEGSGRITCTSRNRCNDQDTRTSYRIGDTWSKKDNRGNLLQCICTGNGRGEWKCERHTS

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 \(c\).

\(|P_{f(3HQN_1)}(2) \setminus P_{f(9JEU_1)}(2)|=188\), \(|P_{f(9JEU_1)}(2) \setminus P_{f(3HQN_1)}(2)|=22\). 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 1:1001100101011011100011011001110000101101110011011010100100000000100100111011101111100011010010111101110011000100011110010000101000010011011001010011111010000000010001000001000011011100101111010001010111001101111011001001101001111010011110010000110010011000011111010111011100111100111000011101110100110010001010010100110111011001110100101001001100110101010011000111001001001110100110001100100001011111000100101110001001110100010000010100110011101001100010000111110110001010010001110100010101000011110
Pair \(Z_2\) Length of longest common subsequence
3HQN_1,9JEU_1 210 3
3HQN_1,1FBR_1 229 3
9JEU_1,1FBR_1 135 2

Newick tree

 
[
	3HQN_1:12.71,
	[
		9JEU_1:67.5,1FBR_1:67.5
	]:53.21
]

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 \((0.85)(0.8)(\frac{617 }{\log_{20} 617}-\frac{118}{\log_{20}118})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 3HQN_1 9JEU_1 189 115
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]