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Parikh vectors
8BQE_1 7QKI_1 3TCX_1 Letter Amino acid
11 44 6 K Lycine
15 5 3 Y Tyrosine
94 27 10 V Valine
203 34 4 A Alanine
76 21 10 L Leucine
31 3 1 F Phenylalanine
75 45 9 S Serine
9 27 4 E Glutamic acid
40 15 1 I Isoleucine
124 49 5 G Glycine
2 0 1 W Tryptophan
0 2 4 C Cysteine
20 19 4 Q Glutamine
53 29 3 D Aspartic acid
3 12 0 H Histidine
4 6 2 M Methionine
8 43 6 P Proline
190 35 6 T Threonine
8 14 2 R Arginine
60 11 4 N Asparagine 

8BQE_1|Chains A, B, C, D, E, F|S-layer protein rsaA|Caulobacter vibrioides NA1000 (565050)
>7QKI_1|Chains A, B, C, D, E, F|Microtubule-associated protein tau|Homo sapiens (9606)
>3TCX_1|Chains A, AA[auth C], C[auth E], E[auth G], G[auth I], I[auth K], K[auth M], M[auth O], O[auth Q], Q[auth S], S[auth U], U[auth W], W[auth Y], Y[auth a]|Intercellular adhesion molecule 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 \(w=f(c)\)
8BQE , Knot 316 1026 0.71 38 191 620 
MAYTTAQLVTAYTNANLGKAPDAATTLTLDAYATQTQTGGLSDAAALTNTLKLVNSTTAVAIQTYQFFTGVAPSAAGLDFLVDSTTNTNDLNDAYYSKFAQENRFINFSINLATGAGAGATAFAAAYTGVSYAQTVATAYDKIIGNAVATAAGVDVAAAVAFLSRQANIDYLTAFVRANTPFTAAADIDLAVKAALIGTILNAATVSGIGGYATATAAMINDLSDGALSTDNAAGVNLFTAYPSSGVSGSTLSLTTGTDTLTGTANNDTFVAGEVAGAATLTVGDTLSGGAGTDVLNWVQAAAVTALPTGVTISGIETMNVTSGAAITLNTSSGVTGLTALNTNTSGAAQTVTAGAGQNLTATTAAQAANNVAVDGGANVTVASTGVTSGTTTVGANSAASGTVSVSVANSSTTTTGAIAVTGGTAVTVAQTAGNAVNTTLTQADVTVTGNSSTTAVTVTQTAAATAGATVAGRVNGAVTITDSAAASATTAGKIATVTLGSFGAATIDSSALTTVNLSGTGTSLGIGRGALTATPTANTLTLNVNGLTTTGAITDSEAAADDGFTTINIAGSTASSTIASLVAADATTLNISGDARVTITSHTAAALTGITVTNSVGATLGAELATGLVFTGGAGADSILLGATTKAIVMGAGDDTVTVSSATLGAGGSVNGGDGTDVLVANVNGSSFSADPAFGGFETLRVAGAAAQGSHNANGFTALQLGATAGATTFTNVAVNVGLTVLAAPTGTTTVTLANATGTSDVFNLTLSSSAALAAGTVALAGVETVNIAATDTNTTAHVDTLTLQATSAKSIVVTGNAGLNLTNTGNTAVTSFDASAVTGTGSAVTFVSANTTVGEVVTIRGGAGADSLTGSATANDTIIGGAGADTLVYTGGTDTFTGGTGADIFDINAIGTSTAFVTITDAAVGDKLDLVGISTNGAIADGAFGAAVTLGAAATLAQYLDAAAAGDGSGTSVAKWFQFGGDTYVVVDSSAGATFVSGADAVIKLTGLVTLTTSAFATEVLTLA
7QKI , Knot 167 441 0.76 38 190 383 
MAEPRQEFEVMEDHAGTYGLGDRKDQGGYTMHQDQEGDTDAGLKESPLQTPTEDGSEEPGSETSDAKSTPTAEDVTAPLVDEGAPGKQAAAQPHTEIPEGTTAEEAGIGDTPSLEDEAAGHVTQARMVSKSKDGTGSDDKKAKGADGKTKIATPRGAAPPGQKGQANATRIPAKTPPAPKTPPSSGEPPKSGDRSGYSSPGSPGTPGSRSRTPSLPTPPTREPKKVAVVRTPPKSPSSAKSRLQTAPVPMPDLKNVKSKIGSTENLKHQPGGGKVQIINKKLDLSNVQSKCGSKDNIKHVPGGGSVQIVYKPVDLSKVTSKCGSLGNIHHKPGGGQVEVKSEKLDFKDRVQSKIGSLDNITHVPGGGNKKIETHKLTFRENAKAKTDHGAEIVYKSPVVSGDTSPRHLSNVSSTGSIDMVDSPQLATLADEVSASLAKQGL
3TCX , Knot 44 85 0.76 38 74 83 
MVSVSPSKVTLPRGGSVLVTCSASCDQPKLLGIETPLVKKELLLPGNNRKVYELSNVQEDSQVMCYANCPDGQSTAKAFLTVYWT

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 \(c\).

\(|P_{f(8BQE_1)}(2) \setminus P_{f(7QKI_1)}(2)|=74\), \(|P_{f(7QKI_1)}(2) \setminus P_{f(8BQE_1)}(2)|=73\). 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 1:110001011010001011011011001010101000001110011110001011000011110000110111101111011100000000100100001100001101010110111111011111001100100110100011101110111101111111100010100101110100110111010111011111011011010111101010111100100111000011110110101001101001010010001010100001111011111010110010111100110110111101110110101100101001111010000110110110000011100101111001010011011001110111010110011001000111001101010101100000001111101101101100110110001001010101000001101000111011101110101110100011101001101101011011110100011001010101001111011101010100101010110001110000111001100101110010001101111010010101010101000011110110100011101110110111101111100111110001111111000101001011111010110100111101010010101111110010111111010001011011011101110010011101110111110100010110101000110101000111111011111100101110000001010010101001001110101110100010011001010110101011011010001101101011111001010101000111111100110011000101101101101011100011101001111001011110001111011111110111110110010111110101001101101110001110001110110110111010111010001110011011
Pair \(Z_2\) Length of longest common subsequence
8BQE_1,7QKI_1 147 4
8BQE_1,3TCX_1 173 4
7QKI_1,3TCX_1 158 5

Newick tree

 
[
	3TCX_1:85.72,
	[
		8BQE_1:73.5,7QKI_1:73.5
	]:12.22
]

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 \((0.85)(0.8)(\frac{1467 }{\log_{20} 1467}-\frac{441}{\log_{20}441})=262.\)
Status Protein1 Protein2 d d1/2
Query variables 8BQE_1 7QKI_1 293 215
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]