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Parikh vectors
8GHB_1 6IFZ_1 8HNA_1 Letter Amino acid
26 50 4 F Phenylalanine
43 8 20 P Proline
42 35 16 R Arginine
17 44 2 N Asparagine
38 56 8 D Aspartic acid
17 6 36 C Cysteine
83 71 3 L Leucine
17 12 0 M Methionine
30 53 13 S Serine
30 37 6 T Threonine
17 43 1 Y Tyrosine
34 31 6 V Valine
44 49 12 E Glutamic acid
29 49 2 I Isoleucine
52 68 17 A Alanine
24 14 13 H Histidine
27 57 6 K Lycine
19 6 9 W Tryptophan
40 29 14 Q Glutamine
39 40 22 G Glycine 

8GHB_1|Chain A|Polyunsaturated fatty acid lipoxygenase ALOX12|Homo sapiens (9606)
>6IFZ_1|Chain A|Type III-A CRISPR-associated protein Csm1|Streptococcus thermophilus ND03 (767463)
>8HNA_1|Chain A[auth B]|Scavenger receptor class F member 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 \(w=f(c)\)
8GHB , Knot 260 668 0.84 40 294 614 
HHHHHHGRYRIRVATGAWLFSGSYNRVQLWLVGTRGEAELELQLRPARGEEEEFDHDVAEDLGLLQFVRLRKHHWLVDDAWFCDRITVQGPGACAEVAFPCYRWVQGEDILSLPEGTARLPGDNALDMFQKHREKELKDRQQIYCWATWKEGLPLTIAADRKDDLPPNMRFHEEKRLDFEWTLKAGALEMALKRVYTLLSSWNCLEDFDQIFWGQKSALAEKVRQCWQDDELFSYQFLNGANPMLLRRSTSLPSRLVLPSGMEELQAQLEKELQNGSLFEADFILLDGIPANVIRGEKQYLAAPLVMLKMEPNGKLQPMVIQIQPPSPSSPTPTLFLPSDPPLAWLLAKSWVRNSDFQLHEIQYHLLNTHLVAEVIAVATMRCLPGLHPIFKFLIPHIRYTMEINTRARTQLISDGGIFDKAVSTGGGGHVQLLRRAAAQLTYCSLCPPDDLADRGLLGLPGALYAHDALRLWEIIARYVEGIVHLFYQRDDIVKGDPELQAWCREITEVGLCQAQDRGFPVSFQSQSQLCHFLTMCVFTCTAQHAAINQGQLDWYAWVPNAPCTMRMPPPTTKEDVTMATVMGSLPDVRQACLQMAISWHLSRRQPDMVPLGHHKEKYFSGPKPKAVLNQFRTDLEKLEKEITARNEQLDWPYEYLKPSCIENSVTI
6IFZ , Knot 289 758 0.84 40 271 677 
MKKEKIDLFYGALLADIGKVIQRATGERKKHALVGADWFDEIADNQVISDQIRYHMANYQSDKLGNDHLAYITYIADNIASGVDRRQSNEESDEDTSAKIWDTYTNQADIFNVFGAQTDKRYFKPTVLNLKSKPNFASATYEPFSKGDYAAIATRIKNELAEFEFNQVQIDSLLNLFEATLSFVPSSTNTKEIADISLADHSRLTAAFALAIYDYLEDKGRHNYKEDLFTKVSAFYEEEAFLLASFDLSGIQDFIYNINIATNGAAKQLKARSLYLDFMSEYIADSLLDKLGLNRANMLYVGGGHAYFVLANTEKTVETLVQFEKDFNQFLLANFQTRLYVAFGWGSFAAKDIMSELNSPESYRQVYQKASRMISKKKISRYDYQTLMLLNRGGKSSERECEICHSVENLVSYHDQKVCDICRGLYQFSKEIAHDHFIITENEGLPIGPNACLKGVAFEKLSQEAFSRVYVKNDYKAGTVKATHVFVGDYQCDEIYNYAALSKNENGLGIKRLAVVRLDVDDLGAAFMAGFSQQGNGQYSTLSRSATFSRSMSLFFKVYINQFASDKKLSIIYAGGDDVFAIGSWQDIIAFTVELRENFIKWTNGKLTLSAGIGLFADKTPISLMAHQTGELEEAAKGNEKDSISLFSSDYTFKFDRFITNVYDDKLEQIRYFFNHQDERGKNFIYKLIELLRNHDRMNMARLAYYLTRLEELTRETDRDKFKTFKNLFYSWYTNKNDKDRKEAELALLLYIYEIRKD
8HNA , Knot 90 210 0.76 38 112 179 
SELDPKGQHVCVASSPSAELQCCAGWRQKDQECTIPICEGPDACQKDEVCVKPGLCRCKPGFFGAHCSSRCPGQYWGPDCRESCPCHPHGQCEPATGACQCQADRWGARCEFPCACGPHGRCDPATGVCHCEPGWWSSTCRRPCQCNTAAARCEQATGACVCKPGWWGRRCSFRCNCHGSPCEQDSGRCACRPGWWGPECQQSRHHHHHH

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 \(c\).

\(|P_{f(8GHB_1)}(2) \setminus P_{f(6IFZ_1)}(2)|=89\), \(|P_{f(6IFZ_1)}(2) \setminus P_{f(8GHB_1)}(2)|=66\). 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 1:00000010001011011111010000101111100101010101011010000100011001111011010000111001110001010111101011110001101001101101010111001101100000001000001001101001111011100000111010100000101010101111011100100110010010010011110001110010001000011000110110111100000110011110110010101000100101101011110111101101000011111111010101010111101011010010101111001111111100110000101001000110001110111110100111101110111101000101000100011001111001100111101011001110100001011001100111111111010011011011100101110110000011010101011000100111001000111101000001001101011000100111001010101111011001011110000010110111011010010101110101000010111110000001011010111001000100100010100001011000101001000101
Pair \(Z_2\) Length of longest common subsequence
8GHB_1,6IFZ_1 155 4
8GHB_1,8HNA_1 230 6
6IFZ_1,8HNA_1 241 3

Newick tree

 
[
	8HNA_1:12.43,
	[
		8GHB_1:77.5,6IFZ_1:77.5
	]:50.93
]

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 \((0.85)(0.8)(\frac{1426 }{\log_{20} 1426}-\frac{668}{\log_{20}668})=190.\)
Status Protein1 Protein2 d d1/2
Query variables 8GHB_1 6IFZ_1 241 229.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]