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Parikh vectors
7ZHU_1 3OTI_1 7YVW_1 Letter Amino acid
25 27 0 P Proline
23 11 0 Q Glutamine
40 24 0 E Glutamic acid
40 20 0 I Isoleucine
26 11 0 F Phenylalanine
6 6 0 W Tryptophan
22 4 0 N Asparagine
39 25 0 D Aspartic acid
46 37 0 L Leucine
26 22 2 T Threonine
8 8 0 M Methionine
32 17 0 S Serine
36 41 0 V Valine
43 49 4 A Alanine
31 31 0 R Arginine
13 21 0 H Histidine
38 4 0 K Lycine
11 1 2 C Cysteine
37 35 3 G Glycine
20 4 0 Y Tyrosine 

7ZHU_1|Chains A, B|Glucose-6-phosphate 1-dehydrogenase|Leishmania donovani (5661)
>3OTI_1|Chains A, B|CalG3|Micromonospora echinospora (1877)
>7YVW_1|Chain A|DNA (5'-D(*CP*TP*AP*AP*CP*GP*GP*AP*AP*TP*G)-3')|synthetic construct (32630)
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 \(w=f(c)\)
7ZHU , Knot 230 562 0.86 40 279 537 
MSEEQSHADQDAYVADVDGILDVLRAQVLERKPDDIFQFISKSALSLQKDRGAESCDRINCKVKDEQKSRALTIIVFGASGDLAKKKTFPALFDLYCGGLLPPEVNIIGYARTKVDDVEKWKHETLMKYFSNLSERGCHAEDFLKHISYFCGAYDSVDDFKRLDAVIREKENAFKGPEKGGNRLFYLALPPSVFASVCESIHKGAMPQEVGGWVRVIIEKPFGRDTKSSAELSQALEPFFDESQLYRIDHYLGKEMVQNIITTRFANRIFSAVWNASNIACVQITFKETIGTEGRGGYFDNIGIIRDVMQNHLTQILALLAMEKPRSLDAECIRDEKVSVLKCIEPITKENCVLGQYTASADGSIPGYLEDVTVPEGSTCPTFAVMRLNINNDRWAGVPFILKAGKAVEQKYVAIRIQFRDEVHPYGEATQRNELVIRAQPSEAMYVKITTKVPGLSGDLRQTHQTELDLTYHTRYDVRLPDAYESLINDALLGNSTNFVRKDELDVAWRIFTPLLHQIDSGEIKPIPYQAGTRGPKEADEFIANNGFKHQKGYHWLPSNKL
3OTI , Knot 161 398 0.80 40 195 362 
GHHHHHHHHHHSSGHIEGRHMRVLFVSSPGIGHLFPLIQLAWGFRTAGHDVLIAVAEHADRAAAAGLEVVDVAPDYSAVKVFEQVAKDNPRFAETVATRPAIDLEEWGVQIAAVNRPLVDGTMALVDDYRPDLVVYEQGATVGLLAADRAGVPAVQRNQSAWRTRGMHRSIASFLTDLMDKHQVSLPEPVATIESFPPSLLLEAEPEGWFMRWVPYGGGAVLGDRLPPVPARPEVAITMGTIELQAFGIGAVEPIIAAAGEVDADFVLALGDLDISPLGTLPRNVRAVGWTPLHTLLRTCTAVVHHGGGGTVMTAIDAGIPQLLAPDPRDQFQHTAREAVSRRGIGLVSTSDKVDADLLRRLIGDESLRTAAREVREEMVALPTPAETVRRIVERISG
7YVW , Knot 8 11 0.58 8 9 9 
CTAACGGAATG

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 \(c\).

\(|P_{f(7ZHU_1)}(2) \setminus P_{f(3OTI_1)}(2)|=128\), \(|P_{f(3OTI_1)}(2) \setminus P_{f(7ZHU_1)}(2)|=44\). 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 1:1000000100010110101110110101100010011011000110100001100000100010000000110111111010110000111110100111111010111010001001001000011001001000100100110010010110001001001011100000110110011001101111101110100010011110011111011100111000000101001101110000100100011001100110001100110111010011010101000110010110100111100110001001111111001001010010000101100101100000111000101010111010010110100010111101010000111111110110110000111010100010101010000011101010011010100011110101000000010100000001011010001100111100001100001011101101110010010101110011001100100111001100001001110001
Pair \(Z_2\) Length of longest common subsequence
7ZHU_1,3OTI_1 172 4
7ZHU_1,7YVW_1 276 3
3OTI_1,7YVW_1 192 3

Newick tree

 
[
	7YVW_1:12.96,
	[
		7ZHU_1:86,3OTI_1:86
	]:41.96
]

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 \((0.85)(0.8)(\frac{960 }{\log_{20} 960}-\frac{398}{\log_{20}398})=149.\)
Status Protein1 Protein2 d d1/2
Query variables 7ZHU_1 3OTI_1 196 162.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]