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Parikh vectors
5BRY_1 8GDI_1 7WAA_1 Letter Amino acid
4 33 11 R Arginine
3 11 20 N Asparagine
0 3 6 C Cysteine
5 10 11 Q Glutamine
7 8 19 K Lycine
5 41 30 A Alanine
4 30 13 E Glutamic acid
15 24 11 I Isoleucine
1 5 14 Y Tyrosine
13 30 24 G Glycine
1 16 10 H Histidine
10 38 25 L Leucine
6 26 13 P Proline
1 23 20 S Serine
8 21 25 T Threonine
6 30 24 V Valine
4 33 30 D Aspartic acid
2 10 11 M Methionine
2 21 16 F Phenylalanine
2 10 3 W Tryptophan 

5BRY_1|Chains A, B|Protease|Human immunodeficiency virus type 1 BH10 (11678)
>8GDI_1|Chain A|Cytochrome P450 124A1|Mycobacterium marinum (1348799)
>7WAA_1|Chains A, B|Probable phosphatidylethanolamine transferase Mcr-1|Escherichia coli (562)
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 \(w=f(c)\)
5BRY , Knot 53 99 0.82 38 77 95 
PQITLWKRPLVTIKIGGQLKEALLDTGADDTVIEEMSLPGRWKPKMIGGIGGFIKVRQYDQIIIEIAGHKAIGTVLVGPTPVNIIGRNLLTQIGATLNF
8GDI , Knot 174 423 0.83 40 219 400 
GLLPRVNGTPPPEVPLADIELGSLEFWGRDDDFRDGAFATLRREAPISFWPPIELAGLTAGKGHWALTKHDDIHFASRHPEIFHSSPNIVIHDQTPELAEYFGSMIVLDDPRHQRLRSIVSRAFTPKVVARIEASVRERAHRLVAAMIENHPDGQADLVSELAGPLPLQIICDMMGIPEEDHEQIFHWTNVILGFGDPDLTTDFDEFLQVSMAIGGYATALADDRRVNHHGDLTTSLVEAEVDGERLSSSEIAMFFILLVVAGNETTRNAISHGMLALSRYPDERAKWWSDFDGLAATAVEEIVRWASPVVYMRRTLSQDVDLRGTKMAAGDKVTLWYCSANRDEEKFADPWTFDVTRNPNPQVGFGGGGAHFCLGANLARREIRVVFDELRRQMPDVVATEEPARLLSQFIHGIKRLPVAWS
7WAA , Knot 147 336 0.84 40 209 322 
HMLEGGSGGSGGSTIYHAKDAVQATKPDMRKPRLVVFVVGETARADHVSFNGYERDTFPQLAKIDGVTNFSNVTSCGTSTAYSVPCMFSYLGADEYDVDTAKYQENVLDTLDRLGVSILWRDNNSDSKGVMDKLPKAQFADYKSATNNAICNTNPYNECRDVGMLVGLDDFVAANNGKDMLIMLHQMGNHGPAYFKRYDEKFAKFTPVCEGNELAKCEHQSLINAYDNALLATDDFIAQSIQWLQTHSNAYDVSMLYVSDHGESLGENGVYLHGMPNAFAPKEQRSVPAFFWTDKQTGITPMATDTVLTHDAITPTLLKLFDVTADKVKDRTAFIR

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 \(c\).

\(|P_{f(5BRY_1)}(2) \setminus P_{f(8GDI_1)}(2)|=23\), \(|P_{f(8GDI_1)}(2) \setminus P_{f(5BRY_1)}(2)|=165\). 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 1:101011001110101110100111001100011001011101010111111111010000011101110011101111101101110011001110101
Pair \(Z_2\) Length of longest common subsequence
5BRY_1,8GDI_1 188 3
5BRY_1,7WAA_1 200 3
8GDI_1,7WAA_1 180 4

Newick tree

 
[
	5BRY_1:99.28,
	[
		8GDI_1:90,7WAA_1:90
	]:9.28
]

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 \((0.85)(0.8)(\frac{522 }{\log_{20} 522}-\frac{99}{\log_{20}99})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 5BRY_1 8GDI_1 159 96.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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