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Parikh vectors
5MSV_1 3BIY_1 6FSF_1 Letter Amino acid
77 25 22 D Aspartic acid
39 11 10 Q Glutamine
63 24 15 P Proline
68 18 22 T Threonine
9 4 2 W Tryptophan
38 13 8 Y Tyrosine
97 23 12 V Valine
84 32 13 E Glutamic acid
26 12 5 H Histidine
44 35 19 K Lycine
147 19 6 A Alanine
18 15 11 N Asparagine
1 7 1 C Cysteine
94 18 10 G Glycine
44 20 12 F Phenylalanine
71 25 15 R Arginine
49 14 19 I Isoleucine
127 30 31 L Leucine
8 8 6 M Methionine
84 27 30 S Serine 

5MSV_1|Chains A, B, C, D|Thioester reductase domain-containing protein|Segniliparus rugosus ATCC BAA-974 (679197)
>3BIY_1|Chain A|Histone acetyltransferase p300|Homo sapiens (9606)
>6FSF_1|Chain A|GTPase-activating protein BEM3|Saccharomyces cerevisiae S288C (559292)
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 \(w=f(c)\)
5MSV , Knot 418 1188 0.83 40 290 997 
MTESQSYETRQARPAGQSLAERVARLVAIDPQAAAAVPDKAVAERATQQGLRLAQRIEAFLSGYGDRPALAQRAFEITKDPITGRAVATLLPKFETVSYRELLERSHAIASELANHAEAPVKAGEFIATIGFTSTDYTSLDIAGVLLGLTSVPLQTGATTDTLKAIAEETAPAVFGASVEHLDNAVTTALATPSVRRLLVFDYRQGVDEDREAVEAARSRLAEAGSAVLVDTLDEVIARGRALPRVALPPATDAGDDSLSLLIYTSGSTGTPKGAMYPERNVAQFWGGIWHNAFDDGDSAPDVPDIMVNFMPLSHVAGRIGLMGTLSSGGTTYFIAKSDLSTFFEDYSLARPTKLFFVPRICEMIYQHYQSELDRIGAADGSPQAEAIKTELREKLLGGRVLTAGSGSAPMSPELTAFIESVLQVHLVDGYGSTEAGPVWRDRKLVKPPVTEHKLIDVPELGYFSTDSPYPRGELAIKTQTILPGYYKRPETTAEVFDEDGFYLTGDVVAEVAPEEFVYVDRRKNVLKLSQGEFVALSKLEAAYGTSPLVRQISVYGSSQRSYLLAVVVPTPEALAKYGDGEAVKSALGDSLQKIAREEGLQSYEVPRDFIIETDPFTIENGILSDAGKTLRPKVKARYGERLEALYAQLAETQAGELRSIRVGAGERPVIETVQRAAAALLGASAAEVDPEAHFSDLGGDSLSALTYSNFLHEIFQVEVPVSVIVSAANNLRSVAAHIEKERSSGSDRPTFASVHGAGATTIRASDLKLEKFLDAQTLAAAPSLPRPASEVRTVLLTGSNGWLGRFLALAWLERLVPQGGKVVVIVRGKDDKAAKARLDSVFESGDPALLAHYEDLADKGLEVLAGDFSDADLGLRKADWDRLADEVDLIVHSGALVNHVLPYSQLFGPNVVGTAEVAKLALTKRLKPVTYLSTVAVAVGVEPSAFEEDGDIRDVSAVRSIDEGYANGYGNSKWAGEVLLREAYEHAGLPVRVFRSDMILAHRKYTGQLNVPDQFTRLILSLLATGIAPKSFYQLDATGGRQRAHYDGIPVDFTAEAITTLGLAGSDGYHSFDVFNPHHDGVGLDEFVDWLVEAGHPISRVDDYAEWLSRFETSLRGLPEAQRQHSVLPLLHAFAQPAPAIDGSPFQTKNFQSSVQEAKVGAEHDIPHLDKALIVKYAEDIKQLGLL
3BIY , Knot 160 380 0.83 40 235 363 
KFSAKRLPSTRLGTFLENRVNDFLRRQNHPESGEVTVRVVHASDKTVEVKPGMKARFVDSGEMAESFPYRTKALFAFEEIDGVDLCFFGMHVQEYGSDCPPPNQRRVYISYLDSVHFFRPKCLRTAVYHEILIGYLEYVKKLGYTTGHIWACPPSEGDDYIFHCHPPDQKIPKPKRLQEWYKKMLDKAVSERIVHDYKDIFKQATEDRLTSAKELPYFEGDFWPNVLEESIKELEQEEEERKREENTSNESTDVTKGDSKNAKKKNNKKTSKNKSSLSRGNKKKPGMPNVSNDLSQKLYATMEKHKEVFFVIRLIAGPAANSLPPIVDPDPLIPCDLMDGRDAFLTLARDRHLEFSSLRRAQWSTGCMLVELHTQSQDRF
6FSF , Knot 119 269 0.82 40 166 257 
GHMKSDIPLFVQPEDFGTIQIEVLSTLYRDNEDDLSILIAIIDRKSGKEMFKFSKSIHKVRELDVYMKSHVPDLPLPTLPDRQLFQTLSPTKVDTRKNILNQYYTSIFSVPEFPKNVGLKIAQFISTDTVMTPPMMDDNVKDGSLLLRRPKTLTGNSTWRVRYGILRDDVLQLFDKNQLTETIKLRQSSIELIPNLPEDRFGTRNGFLITEHKKSGLSTSTKYYICTETSKERELWLSAFSDYIDPSQSLSLSSSRNANDTDSASHLSA

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 \(c\).

\(|P_{f(5MSV_1)}(2) \setminus P_{f(3BIY_1)}(2)|=99\), \(|P_{f(3BIY_1)}(2) \setminus P_{f(5MSV_1)}(2)|=44\). 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 1:100000000001011100110011011110101111110011100100011011001011101010011110011010001101011101110100100001100001110011001011101101110111000000010111111110011100110000101110001111111010010011001110101001111000011000001101100011011011110010011101011101111110011000101110001001010111010001101111110011001001101101110111100111011111010011000111000100110000110100111110100110000000100111101010101100010001111011011010111010101110011010110101000111110000110111000011011011010000101010111000011110000100010110001101010111011100110100000110100101111001011010011100101010000001111111010111001010110011100100110001100001100111000110100111001100101010100100101101011000110100101111001110010011111111011010101010011100101100001100110101110111011001001110100000010001011010111100101001010011010011111011011001001110100111101111111001110110111110100001101010011001011111000011001101111010010111001010011001011100111100111000111101110101101110001011001001111111010110001010010110010010101010001110111001000111110110001111000001010110010011101110111100100101011000100011110101011001111100100010110100011110011011101101100100010110010001011101000001111101110111110101100001000100101110001101001111001001001111
Pair \(Z_2\) Length of longest common subsequence
5MSV_1,3BIY_1 143 4
5MSV_1,6FSF_1 166 4
3BIY_1,6FSF_1 177 3

Newick tree

 
[
	6FSF_1:90.05,
	[
		5MSV_1:71.5,3BIY_1:71.5
	]:18.55
]

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 \((0.85)(0.8)(\frac{1568 }{\log_{20} 1568}-\frac{380}{\log_{20}380})=303.\)
Status Protein1 Protein2 d d1/2
Query variables 5MSV_1 3BIY_1 377 249.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]