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Parikh vectors
8HOG_1 8CMU_1 6AJC_1 Letter Amino acid
16 45 20 R Arginine
11 47 27 D Aspartic acid
1 9 5 C Cysteine
9 32 18 F Phenylalanine
9 45 26 T Threonine
13 71 33 V Valine
4 26 13 Q Glutamine
2 38 35 K Lycine
4 33 12 P Proline
7 29 17 Y Tyrosine
7 40 13 N Asparagine
5 38 24 I Isoleucine
11 48 25 L Leucine
6 20 17 M Methionine
5 15 7 W Tryptophan
9 37 29 A Alanine
16 49 27 E Glutamic acid
13 50 32 G Glycine
5 14 12 H Histidine
9 46 21 S Serine 

8HOG_1|Chain A|Apoptosis regulator Bcl-2|Homo sapiens (9606)
>8CMU_1|Chains A, B|Coagulation factor XIII A chain|Homo sapiens (9606)
>6AJC_1|Chains A, B, C, D, E, F, G, H|Isocitrate dehydrogenase [NADP]|Trypanosoma cruzi (strain CL Brener) (353153)
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 \(w=f(c)\)
8HOG , Knot 79 162 0.82 40 119 153 
SRTGYDNREIVMKYIHYKLSQRGYEWDAGDDVEENRTEAPEGTESEVVHLTLRQAGDDFSRRYRRDFAEMSSQLHLTPFTARGRFATVVEELFRDGVNWGRIVAFFEFGGVMCVESVNREMSPLVDNIALWMTEYLNRHLHTWIQDNGGWDAFVELYGPSMR
8CMU , Knot 287 732 0.86 40 298 675 
MSETSRTAFGGRRAVPPNNSNAAEDDLPTVELQGVVPRGVNLQEFLNVTSVHLFKERWDTNKVDHHTDKYENNKLIVRRGQSFYVQIDFSRPYDPRRDLFRVEYVIGRYPQENKGTYIPVPIVSELQSGKWGAKIVMREDRSVRLSIQSSPKCIVGKFRMYVAVWTPYGVLRTSRNPETDTYILFNPWCEDDAVYLDNEKEREEYVLNDIGVIFYGEVNDIKTRSWSYGQFEDGILDTCLYVMDRAQMDLSGRGNPIKVSRVGSAMVNAKDDEGVLVGSWDNIYAYGVPPSAWTGSVDILLEYRSSENPVRYGQCWVFAGVFNTFLRCLGIPARIVTNYFSAHDNDANLQMDIFLEEDGNVNSKLTKDSVWNYHCWNEAWMTRPDLPVGFGGWQAVDSTPQENSDGMYRCGPASVQAIKHGHVCFQFDAPFVFAEVNSDLIYITAKKDGTHVVENVDATHIGKLIVTKQIGGDGMMDITDTYKFQEGQEEERLALETALMYGAKKPLNTEGVMKSRSNVDMDFEVENAVLGKDFKLSITFRNNSHNRYTITAYLSANITFYTGVPKAEFKKETFDVTLEPLSFKKEAVLIQAGEYMGQLLEQASLHFFVTARINETRDVLAKQKSTVLTIPEIIIKVRGTQVVGSDMTVTVEFTNPLKETLRNVWVHLDGPGVTRPMKKMFREIRPNSTVQWEEVCRPWVSGHRKLIASMSSDSLRHVYGELDVQIQRRPSM
6AJC , Knot 178 413 0.86 40 240 398 
SQKIKVAGTVVELDGDEMTRVIWKMIKEELIFPFLDVPIEYYDLGMENRDKTDDQVTVDAAHAIKKHGVGVKCATITPDEARVREFNLKQMWKSPNGTIRNILGGTVFREPIMCKNVPRLVTTWKHPIVIGRHAFGDQYRATDLVVNGPGTFEIHFVPESGGAAQVQKVFDFKSGGVLMGMYNTDESIKDFAKSCFEYALSKKWPLYLSTKNTILKRYDGRFKDIFAEMYKASYEADYKKAGIWYEHRLIDDMVAYAMKSEGGYVWACKNYDGDVQSDSVAQGFGSLGLMTSVLMSPDGRTVEAEAAHGTVTRHYRQHQKGEETSTNPVASIFAWTRGLMHRGKLDQNEKLVQFSMLLEKVVVSTIEAGFMTKDLAICIKGMNHVTRSDYLNTQEFIHKLADEMRKAYERSKI

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 \(c\).

\(|P_{f(8HOG_1)}(2) \setminus P_{f(8CMU_1)}(2)|=17\), \(|P_{f(8CMU_1)}(2) \setminus P_{f(8HOG_1)}(2)|=196\). 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 1:000100000111001000100010010110010000001101000011010100110010000000110100010101101010110110011001101101111101111101001000101110011111000100010011000111011101011010
Pair \(Z_2\) Length of longest common subsequence
8HOG_1,8CMU_1 213 4
8HOG_1,6AJC_1 201 4
8CMU_1,6AJC_1 160 4

Newick tree

 
[
	8HOG_1:11.27,
	[
		6AJC_1:80,8CMU_1:80
	]:30.27
]

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 \((0.85)(0.8)(\frac{894 }{\log_{20} 894}-\frac{162}{\log_{20}162})=203.\)
Status Protein1 Protein2 d d1/2
Query variables 8HOG_1 8CMU_1 263 158.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]