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Parikh vectors
1USH_1 7RPN_1 1DFL_1 Letter Amino acid
35 23 62 E Glutamic acid
50 21 52 G Glycine
41 19 79 K Lycine
21 7 32 Y Tyrosine
40 18 45 V Valine
20 5 38 R Arginine
27 14 46 N Asparagine
16 4 24 M Methionine
21 16 47 S Serine
7 2 8 W Tryptophan
44 18 70 L Leucine
19 8 41 F Phenylalanine
26 8 33 P Proline
48 35 55 A Alanine
38 10 42 D Aspartic acid
28 5 37 Q Glutamine
10 3 15 H Histidine
28 21 54 I Isoleucine
29 10 36 T Threonine
2 5 15 C Cysteine 

1USH_1|Chain A|5'-NUCLEOTIDASE|Escherichia coli (83333)
>7RPN_1|Chain A|Triosephosphate isomerase|Bacteroides thetaiotaomicron (818)
>1DFL_1|Chains A, D[auth B]|MYOSIN HEAD|Argopecten irradians (31199)
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 \(w=f(c)\)
1USH , Knot 224 550 0.85 40 266 514 
MKLLQRGVALALLTTFTLASETALAYEQDKTYKITVLHTNDHHGHFWRNEYGEYGLAAQKTLVDGIRKEVAAEGGSVLLLSGGDINTGVPESDLQDAEPDFRGMNLVGYDAMAIGNHEFDNPLTVLRQQEKWAKFPLLSANIYQKSTGERLFKPWALFKRQDLKIAVIGLTTDDTAKIGNPEYFTDIEFRKPADEAKLVIQELQQTEKPDIIIAATHMGHYDNGEHGSNAPGDVEMARALPAGSLAMIVGGHSQDPVCMAAENKKQVDYVPGTPCKPDQQNGIWIVQAHEWGKYVGRADFEFRNGEMKMVNYQLIPVNLKKKVTWEDGKSERVLYTPEIAENQQMISLLSPFQNKGKAQLEVKIGETNGRLEGDRDKVRFVQTNMGRLILAAQMDRTGADFAVMSGGGIRDSIEAGDISYKNVLKVQPFGNVVVYADMTGKEVIDYLTAVAQMKPDSGAYPQFANVSFVAKDGKLNDLKIKGEPVDPAKTYRMATLNFNATGGDGYPRLDNKPGYVNTGFIDAEVLKAYIQKSSPLDVSVYEPKGEVSWQ
7RPN , Knot 111 252 0.81 40 157 240 
MRKNIVAGNWKMNKTLQEGIALAKELNEALANEKPNCDVIICTPFIHLASVTPLVDAAKIGVGAENCADKASGAYTGEVSAEMVASTGAKYVILGHSERRAYYGETVAILEEKVKLALANGLTPIFCIGEVLEEREANKQNEVVAAQMESVFSLSAEDFSKIILAYEPVWAIGTGKTASPEQAQEIHAFIRSIVADKYGKEIADNTSILYGGSCKPSNAKELFSNPDVDGGLIGGAALKVSDFKGIIDAFNA
1DFL , Knot 312 831 0.84 40 305 756 
FSDPDFQYLAVDRKKLMKEQTAAFDGKKNCWVPDEKEGFASAEIQSSKGDEITVKIVADSSTRTVKKDDIQSMNPPKFEKLEDMANMTYLNEASVLYNLRSRYTSGLIYTYSGLFCIAVNPYRRLPIYTDSVIAKYRGKRKTEIPPHLFSVADNAYQNMVTDRENQSCLITGESGAGKTENTKKVIMYLAKVACAVKKKDEEASDKKEGSLEDQIIQANPVLEAYGNAKTTRNNNSSRFGKFIRIHFGPTGKIAGADIETYLLEKSRVTYQQSAERNYHIFYQICSNAIPELNDVMLVTPDSGLYSFINQGCLTVDNIDDVEEFKLCDEAFDILGFTKEEKQSMFKCTASILHMGEMKFKQRPREEQAESDGTAEAEKVAFLCGINAGDLLKALLKPKVKVGTEMVTKGQNMNQVVNSVGALAKSLYDRMFNWLVRRVNKTLDTKAKRNYYIGVLDIAGFEIFDFNSFEQLCINYTNERLQQFFNHHMFILEQEEYKKEGIAWEFIDFGMDLQMCIDLIEKPMGILSILEEECMFPKADDKSFQDKLYQNHMGKNRMFTKPGKPTRPNQGPAHFELHHYAGNVPYSITGWLEKNKDPINENVVALLGASKEPLVAELFKAPEEPAGGGKKKKGKSSAFQTISAVHRESLNKLMKNLYSTHPHFVRCIIPNELKQPGLVDAELVLHQLQCNGVLEGIRICRKGFPSRLIYSEFKQRYSILAPNAIPQGFVDGKTVSEKILAGLQMDPAEYRLGTTKVFFKAGVLGNLEEMRDERLSKIISMFQAHIRGYLIRKAYKKLQDQRIGLSVIQRNIRKWLVLRNWQWWKLYSKVKP

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 \(c\).

\(|P_{f(1USH_1)}(2) \setminus P_{f(7RPN_1)}(2)|=140\), \(|P_{f(7RPN_1)}(2) \setminus P_{f(1USH_1)}(2)|=31\). 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 1:1011001111111001011000111000000001011000000101100001001111000110110001110110111101101001110001001010101101110011111000100110110000011011110101000001001101111100001011111100000101101001001010011001011100100000101111100110000100100111010110111110111111100001101110000010011101001000011111010011001101010100101011000111101000101001000011001011000011011011000101010101100010101000010110001101111101000110111101111000101101000011010111011101010100110010111010100110101101011100101001010101101100001101010101101010100011010011101011010100001101010010101010
Pair \(Z_2\) Length of longest common subsequence
1USH_1,7RPN_1 171 4
1USH_1,1DFL_1 133 4
7RPN_1,1DFL_1 178 4

Newick tree

 
[
	7RPN_1:93.16,
	[
		1USH_1:66.5,1DFL_1:66.5
	]:26.66
]

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 \((0.85)(0.8)(\frac{802 }{\log_{20} 802}-\frac{252}{\log_{20}252})=151.\)
Status Protein1 Protein2 d d1/2
Query variables 1USH_1 7RPN_1 190 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]