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Parikh vectors
6QGB_1 3MLH_1 7ZBZ_1 Letter Amino acid
23 12 38 N Asparagine
37 14 41 D Aspartic acid
66 17 32 G Glycine
23 9 29 F Phenylalanine
14 6 6 W Tryptophan
10 4 12 C Cysteine
19 14 33 I Isoleucine
31 9 41 T Threonine
28 8 39 R Arginine
14 17 68 K Lycine
20 2 18 M Methionine
41 27 51 S Serine
17 12 35 Y Tyrosine
91 15 46 A Alanine
25 5 53 Q Glutamine
15 14 62 E Glutamic acid
14 11 15 H Histidine
38 14 93 L Leucine
44 12 17 P Proline
26 13 47 V Valine 

6QGB_1|Chains A, B, C, D, E, F|Mono(2-hydroxyethyl) terephthalate hydrolase|Ideonella sakaiensis (1547922)
>3MLH_1|Chains A, B|Hemagglutinin|Influenza A virus (644887)
>7ZBZ_1|Chain A[auth C]|Cullin-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 \(w=f(c)\)
6QGB , Knot 229 596 0.81 40 254 537 
MNHKVHHHHHHMGGGSTPLPLPQQQPPQQEPPPPPVPLASRAACEALKDGNGDMVWPNAATVVEVAAWRDAAPATASAAALPEHCEVSGAIAKRTGIDGYPYEIKFRLRMPAEWNGRFFMEGGSGTNGSLSAATGSIGGGQIASALSRNFATIATDGGHDNAVNDNPDALGTVAFGLDPQARLDMGYNSYDQVTQAGKAAVARFYGRAADKSYFIGCSEGGREGMMLSQRFPSHYDGIVAGAPGYQLPKAGISGAWTTQSLAPAAVGLDAQGVPLINKSFSDADLHLLSQAILGTCDALDGLADGIVDNYRACQAAFDPATAANPANGQALQCVGAKTADCLSPVQVTAIKRAMAGPVNSAGTPLYNRWAWDAGMSGLSGTTYNQGWRSWWLGSFNSSANNAQRVSGFSARSWLVDFATPPEPMPMTQVAARMMKFDFDIDPLKIWATSGQFTQSSMDWHGATSTDLAAFRDRGGKMILYHGMSDAAFSALDTADYYERLGAAMPGAAGFARLFLVPGMNHCSGGPGTDRFDMLTPLVAWVERGEAPDQISAWSGTPGYFGVAARTRPLCPYPQIARYKGSGDINTEANFACAAPP
3MLH , Knot 109 235 0.84 40 168 223 
HHHHHHDDDDKGVAPLHLGKCNIAGWILGNPECESLSTASSWSYIVETSSSDNGTCYPGDFIDYEELREQLSSVSSFERFEIFPKTSSWPNHDSNKGVTAACPHAGAKSFYKNLIWLVKKGNSYPKLSKSYINDKGKEVLVLWGIHHPSTSADQQSLYQNADAYVFVGSSRYSKKFKPEIAIRPKVRDQEGRMNYYWTLVEPGDKITFEATGNLVVPRYAFAMERNAGSGIIISD
7ZBZ , Knot 297 776 0.85 40 289 704 
MSSTRSQNPHGLKQIGLDQIWDDLRAGIQQVYTRQSMAKSRYMELYTHVYNYCTSVHQSNQARGAGVPPSKSKKGQTPGGAQFVGLELYKRLKEFLKNYLTNLLKDGEDLMDESVLKFYTQQWEDYRFSSKVLNGICAYLNRHWVRRECDEGRKGIYEIYSLALVTWRDCLFRPLNKQVTNAVLKLIEKERNGETINTRLISGVVQSYVELGLNEDDAFAKGPTLTVYKESFESQFLADTERFYTRESTEFLQQNPVTEYMKKAEARLLEEQRRVQVYLHESTQDELARKCEQVLIEKHLEIFHTEFQNLLDADKNEDLGRMYNLVSRIQDGLGELKKLLETHIHNQGLAAIEKCGEAALNDPKMYVQTVLDVHKKYNALVMSAFNNDAGFVAALDKACGRFINNNAVTKMAQSSSKSPELLARYCDSLLKKSSKNPEEAELEDTLNQVMVVFKYIEDKDVFQKFYAKMLAKRLVHQNSASDDAEASMISKLKQACGFEYTSKLQRMFQDIGVSKDLNEQFKKHLTNSEPLDLDFSIQVLSSGSWPFQQSCTFALPSELERSYQRFTAFYASRHSGRKLTWLYQLSKGELVTNCFKNRYTLQASTFQMAILLQYNTEDAYTVQQLTDSTQIKMDILAQVLQILLKSKLLVLEDENANVDEVELKPDTLIKLYLGYKNKKLRVNINVPMKTEQKQEQETTHKNIEEDRKLLIQAAIVRIMKMRKVLKHQQLLGEVLTQLSSRFKPRVPVIKKCIDILIEKEYLERVDGEKDTYSYLA

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 \(c\).

\(|P_{f(6QGB_1)}(2) \setminus P_{f(3MLH_1)}(2)|=135\), \(|P_{f(3MLH_1)}(2) \setminus P_{f(6QGB_1)}(2)|=49\). 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 1:10001000000111100111110001100011111111100110011001010111101101101111001111010111110000101111000110101001010101110101011101101001010110101111011011000110110011000110001011101111101010101100000010011011110101011000011100011001111000110000111111110011011101110000111111110101111100010010101100111100011011101110000100111011011011010110011100100101101011001111110011011000111011101101000001100111101000100100101101001110110110111100111011010101011011100101000010101100001111000110111001100111011001000001111111111110111111100001111000101101111110010110010110101101111100011010101100010101000101101111
Pair \(Z_2\) Length of longest common subsequence
6QGB_1,3MLH_1 184 6
6QGB_1,7ZBZ_1 165 4
3MLH_1,7ZBZ_1 193 4

Newick tree

 
[
	3MLH_1:97.88,
	[
		6QGB_1:82.5,7ZBZ_1:82.5
	]:15.38
]

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 \((0.85)(0.8)(\frac{831 }{\log_{20} 831}-\frac{235}{\log_{20}235})=164.\)
Status Protein1 Protein2 d d1/2
Query variables 6QGB_1 3MLH_1 201 140.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]