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Parikh vectors
9OPF_1 4MQE_1 1CDG_1 Letter Amino acid
0 19 28 P Proline
4 27 56 T Threonine
5 22 20 R Arginine
2 16 62 N Asparagine
2 27 40 D Aspartic acid
1 5 3 C Cysteine
2 9 13 M Methionine
0 19 32 F Phenylalanine
1 18 33 Y Tyrosine
2 23 49 V Valine
10 29 65 A Alanine
13 28 18 E Glutamic acid
1 29 65 G Glycine
5 25 37 I Isoleucine
10 14 26 Q Glutamine
9 32 38 L Leucine
9 27 29 K Lycine
4 33 47 S Serine
0 11 13 W Tryptophan
1 7 12 H Histidine 

9OPF_1|Chains A, B|Transforming acidic coiled-coil-containing protein 3|Homo sapiens (9606)
>4MQE_1|Chain A|Gamma-aminobutyric acid type B receptor subunit 1|Homo sapiens (9606)
>1CDG_1|Chain A|CYCLODEXTRIN GLYCOSYL-TRANSFERASE|Bacillus circulans (1397)
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 \(w=f(c)\)
9OPF , Knot 42 81 0.76 34 60 78 
TQEGQRYQALKAHAEEKLQLANEEIAQVRSKAQAEALALQASLRKEQMRIQSLEKTVEQKTKENEELTRICDDLISKMEKI
4MQE , Knot 180 420 0.86 40 243 398 
SERRAVYIGALFPMSGGWPGGQACQPAVEMALEDVNSRRDILPDYELKLIHHDSKCDPGQATKYLYELLYNDPIKIILMPGCSSVSTLVAEAARMWNLIVLSYGSSSPALSNRQRFPTFFRTHPSATLHNPTRVKLFEKWGWKKIATIQQTTEVFTSTLDDLEERVKEAGIEITFRQSFFSDPAVPVKNLKRQDARIIVGLFYETEARKVFCEVYKERLFGKKYVWFLIGWYADNWFKIYDPSINCTVDEMTEAVEGHITTEIVMLNPANTRSISNMTSQEFVEKLTKRLKRHPEETGGFQEAPLAYDAIWALALALNKTSGGGGRSGVRLEDFNYNNQTITDQIYRAMNSSSFEGVSGHVVFDASGSRMAWTLIEQLQGGSYKKIGYYDSTKDDLSWSKTDKWIGGSPPADDYKDDDDK
1CDG , Knot 265 686 0.84 40 269 621 
APDTSVSNKQNFSTDVIYQIFTDRFSDGNPANNPTGAAFDGTCTNLRLYCGGDWQGIINKINDGYLTGMGVTAIWISQPVENIYSIINYSGVNNTAYHGYWARDFKKTNPAYGTIADFQNLIAAAHAKNIKVIIDFAPNHTSPASSDQPSFAENGRLYDNGTLLGGYTNDTQNLFHHNGGTDFSTTENGIYKNLYDLADLNHNNSTVDVYLKDAIKMWLDLGIDGIRMDAVKHMPFGWQKSFMAAVNNYKPVFTFGEWFLGVNEVSPENHKFANESGMSLLDFRFAQKVRQVFRDNTDNMYGLKAMLEGSAADYAQVDDQVTFIDNHDMERFHASNANRRKLEQALAFTLTSRGVPAIYYGTEQYMSGGTDPDNRARIPSFSTSTTAYQVIQKLAPLRKCNPAIAYGSTQERWINNDVLIYERKFGSNVAVVAVNRNLNAPASISGLVTSLPQGSYNDVLGGLLNGNTLSVGSGGAASNFTLAAGGTAVWQYTAATATPTIGHVGPMMAKPGVTITIDGRGFGSSKGTVYFGTTAVSGADITSWEDTQIKVKIPAVAGGNYNIKVANAAGTASNVYDNFEVLSGDQVSVRFVVNNATTALGQNVYLTGSVSELGNWDPAKAIGPMYNQVVYQYPNWYYDVSVPAGKTIEFKFLKKQGSTVTWEGGSNHTFTAPSSGTATINVNWQP

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 \(c\).

\(|P_{f(9OPF_1)}(2) \setminus P_{f(4MQE_1)}(2)|=18\), \(|P_{f(4MQE_1)}(2) \setminus P_{f(9OPF_1)}(2)|=201\). 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 1:000100001101010001011000110100010101111010100001010010001000000000100100011001001
Pair \(Z_2\) Length of longest common subsequence
9OPF_1,4MQE_1 219 4
9OPF_1,1CDG_1 243 3
4MQE_1,1CDG_1 148 4

Newick tree

 
[
	9OPF_1:12.52,
	[
		4MQE_1:74,1CDG_1:74
	]:52.52
]

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 \((0.85)(0.8)(\frac{501 }{\log_{20} 501}-\frac{81}{\log_{20}81})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 9OPF_1 4MQE_1 167 97.5
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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