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Parikh vectors
1ZOP_1 2VZM_1 3UFH_1 Letter Amino acid
13 24 75 T Threonine
12 49 96 A Alanine
12 28 41 E Glutamic acid
15 11 26 F Phenylalanine
4 33 44 P Proline
6 8 29 Y Tyrosine
16 32 50 D Aspartic acid
11 21 72 S Serine
0 6 4 W Tryptophan
9 32 78 G Glycine
5 7 44 N Asparagine
1 2 26 C Cysteine
7 8 18 Q Glutamine
4 18 36 H Histidine
15 13 36 I Isoleucine
17 52 57 L Leucine
21 4 55 K Lycine
4 45 20 R Arginine
11 30 57 V Valine
4 13 10 M Methionine 

1ZOP_1|Chains A, B|I-DOMAIN FRAGMENT OF LFA-1|Homo sapiens (9606)
>2VZM_1|Chains A, B|CYTOCHROME P450 MONOOXYGENASE|STREPTOMYCES VENEZUELAE (54571)
>3UFH_1|Chain A|UndA|Shewanella sp. HRCR_06 (1043181)
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 \(w=f(c)\)
1ZOP , Knot 87 187 0.81 38 130 181 
CIKGNVDLVFLFDGSMSLQPDEFQKILDFMKDVMKKLSNTSYQFAAVQFSTSYKTEFDFSDYVKRKDPDALLKHVKHMLLLTNTFGAINYVATEVFREELGARPDATKVLIIITDGEATDSGNIDAAKDIIRYIIGIGKHFQTKESQETLHKFASKPASEFVKILDTFEKLKDLFTELQKKIYVIEG
2VZM , Knot 177 436 0.82 40 213 397 
MGSSHHHHHHSSGLVPRGSHMRRTQQGTTASPPVLDLGALGQDFAADPYPTYARLRAEGPAHRVRTPEGNEVWLVVGYDRARAVLADPRFSKDWRNSTTPLTEAEAALNHNMLESDPPRHTRLRKLVAREFTMRRVELLRPRVQEIVDGLVDAMLAAPDGRADLMESLAWPLPITVISELLGVPEPDRAAFRVWTDAFVFPDDPAQAQTAMAEMSGYLSRLIDSKRGQDGEDLLSALVRTSDEDGSRLTSEELLGMAHILLVAGHETTVNLIANGMYALLSHPDQLAALRADMTLLDGAVEEMLRYEGPVESATYRFPVEPVDLDGTVIPAGDTVLVVLADAHRTPERFPDPHRFDIRRDTAGHLAFGHGIHFCIGAPLARLEARIAVRALLERCPDLALDVSPGELVWYPNPMIRGLKALPIRWRRGREAGRRTG
3UFH , Knot 320 874 0.82 40 283 758 
MSKKLLSVLFGASLAALALSPTAFAADQGGSDGKDGEDGKPGPVGLDISQATTLKATLEDVKIDNGTVSVDIVLTNANGVPVTGLEQYAQINAIGLGIAKLTPESGKGYKTPQWVSYINSVKAADPARSLANYSYTDGKDSAGNPITKEVKFTPGDAIQANIESSCKTTCLTVVDSGVYRYTFQTNLSTLPAIEGLDLTYDPTLIHRITLELQTDGSKDAKLVNSHIDFLPSDNFRVAKETETRTVVDLEANCIKCHSTNYSDTSSTAKPLALHGGRRIGIANCQVCHTSYSKDPETGSPLDMGAMVHAIHKGTYAMVGYSGTAYDFSGTMAKAAAESGYPQYREGKDVSERVTLPVSIGNCQSCHSTDDKGPVDAASFKHHKGLACASCHMSGFNPVDNSEWLTPPEGQKDRGFVGNYFHYYATPEIDGIPGVNLVHVFQNGGCASCHAEQGEEGSAKYHLAKANATKLLRTEYAYKLENGTFDVAKGELTFTVNWHSDVAPHQDPKVKEFWVSLTAFNGTEYTMGPRPSNGTLGRSENRISVNLAKVETNANLTAVPNGSKVTYTLTGIKAVIGTSSVPYKQIVSIGKGFMDGKLLICANSAELDPTMDAAIDCSNTEAPIYEVIVGSNKASFSADASNVTARSIVISEAKCANCHGEKADFSASHALTHAADKPDNSCGTCHSAVPNTAVALADGSCVACHNGAPAHSKKPFERGFDFKVMIHQIHADTRSVRRLTTDAATFPENPANCAACHDKGQLSLATLGNKPAFLASTGEYSPTVAACASCHATTATDSAVIGHFETNGGVYNAAAGTYTPGSETCATCHGEGKSFGVDKVHPVKYKGELKLEGKPIPNPLLGLDSTRTGHHHHHH

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 \(c\).

\(|P_{f(1ZOP_1)}(2) \setminus P_{f(2VZM_1)}(2)|=49\), \(|P_{f(2VZM_1)}(2) \setminus P_{f(1ZOP_1)}(2)|=132\). 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 1:0101010111110101010100100110110011001000000111101000000010100010000101110010011110001111001100110001110101001111100101000101011001100111110010000000010011001100110110010010011001000101101
Pair \(Z_2\) Length of longest common subsequence
1ZOP_1,2VZM_1 181 3
1ZOP_1,3UFH_1 185 4
2VZM_1,3UFH_1 154 6

Newick tree

 
[
	1ZOP_1:95.85,
	[
		2VZM_1:77,3UFH_1:77
	]:18.85
]

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 \((0.85)(0.8)(\frac{623 }{\log_{20} 623}-\frac{187}{\log_{20}187})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 1ZOP_1 2VZM_1 161 114.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]