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Parikh vectors
3EUE_1 8GIS_1 3MCA_1 Letter Amino acid
27 17 33 G Glycine
32 40 53 L Leucine
18 43 42 K Lycine
12 16 23 P Proline
16 15 29 T Threonine
0 5 5 W Tryptophan
11 11 21 Q Glutamine
13 8 3 C Cysteine
20 38 48 E Glutamic acid
11 7 10 H Histidine
18 21 31 I Isoleucine
10 5 9 M Methionine
12 22 20 F Phenylalanine
27 12 36 A Alanine
23 25 59 S Serine
13 14 32 N Asparagine
12 16 39 D Aspartic acid
12 11 15 Y Tyrosine
24 18 52 V Valine
17 20 32 R Arginine 

3EUE_1|Chain A|Uridine phosphorylase 1|Homo sapiens (9606)
>8GIS_1|Chains A, B[auth C]|Cyclic GMP-AMP synthase|Mus musculus (10090)
>3MCA_1|Chain A|Elongation factor 1 alpha-like protein|Schizosaccharomyces pombe (4896)
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 \(w=f(c)\)
3EUE , Knot 141 328 0.83 38 197 308 
MRGSHHHHHHGSPGLQEFMAATGANAEKAESHNDCPVRLLNPNIAKMKEDILYHFNLTTSRHNFPALFGDVKFVCVGGSPSRMKAFIRCVGAELGLDCPGRDYPNICAGTDRYAMYKVGPVLSVSHGMGIPSISIMLHELIKLLYYARCSNVTIIRIGTSGGIGLEPGTVVITEQAVDTCFKAEFEQIVLGKRVIRKTDLNKKLVQELLLCSAELSEFTTVVGNTMCTLDFYEGQGRLDGALCSYTEKDKQAYLEAAYAAGVRNIEMESSVFAAMCSACGLQAAVVCVTLLNRLEGDQISSPRNVLSEYQQRPQRLVSYFIKKKLSKA
8GIS , Knot 156 364 0.84 40 198 344 
GTGPDKLKKVLDKLRLKRKDISEAAETVNKVVERLLRRMQKRESEFKGVEQLNTGSYYEHVKISAPNEFDVMFKLEVPRIELQEYYETGAFYLVKFKRIPRGNPLSHFLEGEVLSATKMLSKFRKIIKEEVKEIKDIDVSVEKEKPGSPAVTLLIRNPEEISVDIILALESKGSWPISTKEGLPIQGWLGTKVRTNLRREPFYLVPKNAKDGNSFQGETWRLSFSHTEKYILNNHGIEKTCCESSGAKCCRKECLKLMKYLLEQLKKEFQELDAFCSYHVKTAIFHMWTQDPQDSQWDPRNLSSCFDKLLAFFLECLRTEKLDHYFIPKFNLFSQELIDRKSKEFLSKKIEYERNNGFPIFDKL
3MCA , Knot 229 592 0.82 40 250 542 
MSRHRDVKNLDLDDYELDEEPGEEELTEEQEEEFRSAVATVRETLLGVPISEKEIADTVWYYYFDVEKSVNYLLQKASSKAGAKEKQNTDSQKEKKQNKSKEALADAKDPLDESSNGIKNLSLNKNDEPAFQTNGEVKMKNSSESDNQPEKKKIKKQNPTDLVSVPEIFEQSNPKPVVHLVVTGHVDSGKSTMLGRIMFELGEINSRSMQKLHNEAANSGKGSFSYAWLLDTTEEERARGVTMDVASTTFESDKKIYEIGDAPGHRDFISGMIAGASSADFAVLVVDSSQNNFERGFLENGQTREHAYLLRALGISEIVVSVNKLDLMSWSEDRFQEIKNIVSDFLIKMVGFKTSNVHFVPISAISGTNLIQKDSSDLYKWYKGPTLLSALDQLVPPEKPYRKPLRLSIDDVYRSPRSVTVTGRVEAGNVQVNQVLYDVSSQEDAYVKNVIRNSDPSSTWAVAGDTVTLQLADIEVNQLRPGDILSNYENPVRRVRSFVAEIQTFDIHGPILSGSTLVLHLGRTVTSVSLKIVTVNNKRSRHIASRKRALVRISFLDGLFPLCLAEECPALGRFILRRSGDTVAAGIVKELC

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 \(c\).

\(|P_{f(3EUE_1)}(2) \setminus P_{f(8GIS_1)}(2)|=87\), \(|P_{f(8GIS_1)}(2) \setminus P_{f(3EUE_1)}(2)|=88\). 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 1:1010000000101110011110110100100000011011010110100011001010000001111110101101110100101110011101110011000101011000011001111101001111101011100110110010000101101100111110110111000110001010100111100110000100011001110010100100111001001010010101011100000000010101101111001010001111100101101111010110010100100100110000001001100110001001
Pair \(Z_2\) Length of longest common subsequence
3EUE_1,8GIS_1 175 4
3EUE_1,3MCA_1 159 3
8GIS_1,3MCA_1 148 4

Newick tree

 
[
	3EUE_1:86.55,
	[
		3MCA_1:74,8GIS_1:74
	]:12.55
]

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 \((0.85)(0.8)(\frac{692 }{\log_{20} 692}-\frac{328}{\log_{20}328})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 3EUE_1 8GIS_1 130 122
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]