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Parikh vectors
1HOP_1 2IRZ_1 3IQF_1 Letter Amino acid
31 22 24 D Aspartic acid
6 7 2 H Histidine
20 23 13 I Isoleucine
22 14 18 K Lycine
12 20 10 F Phenylalanine
16 30 9 S Serine
4 7 0 W Tryptophan
16 19 6 Y Tyrosine
28 20 14 R Arginine
44 30 22 L Leucine
17 22 16 P Proline
33 24 6 T Threonine
13 15 8 N Asparagine
10 16 3 Q Glutamine
46 35 14 G Glycine
9 9 17 M Methionine
32 26 35 A Alanine
4 6 3 C Cysteine
29 26 37 E Glutamic acid
39 34 26 V Valine 

1HOP_1|Chains A, B|ADENYLOSUCCINATE SYNTHETASE|Escherichia coli (562)
>2IRZ_1|Chain A|Beta-secretase 1|Homo sapiens (9606)
>3IQF_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|F420-dependent methylenetetrahydromethanopterin dehydrogenase|Methanopyrus kandleri (2320)
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 \(w=f(c)\)
1HOP , Knot 177 431 0.83 40 218 408 
GNNVVVLGTQWGDEGKGKIVDLLTERAKYVVRYQGGHNAGHTLVINGEKTVLHLIPSGILRENVTSIIGNGVVLSPAALMKEMKELEDRGIPVRERLLLSEACPLILDYHVALDNAREKARGAKAIGTTGRGIGPAYEDKVARRGLRVGDLFDKETFAEKLKEVMEYHNFQLVNYYKAEAVDYQKVLDDTMAVADILTSMVVDVSDLLDQARQRGDFVMFEGAQGTLLDIDHGTYPYVTSSNTTAGGVATGSGLGPRYVDYVLGILKAYSTRVGAGPFPTELFDETGEFLCKQGNEFGATTGRRRRTGWLDTVAVRRAVQLNSLSGFCLTKLDVLDGLKEVKLCVAYRMPDGREVTTTPLAADDWKGVEPIYETMPGWSESTFGVKDRSGLPQAALNYIKRIEELTGVPIDIISTGPDRTETMILRDPFDA
2IRZ , Knot 175 405 0.86 40 234 389 
MLPRETDEEPEEPGRRGSFVEMVDNLRGKSGQGYYVEMTVGSPPQTLNILVDTGSSNFAVGAAPHPFLHRYYQRQLSSTYRDLRKGVYVPYTQGAWAGELGTDLVSIPHGPNVTVRANIAAITESDKFFINGSNWEGILGLAYAEIARPDDSLEPFFDSLVKQTHVPNLFSLQLCGAGFPLNQSEVLASVGGSMIIGGIDHSLYTGSLWYTPIRREWYYEVIIVRVEINGQDLKMDCKEYNYDKSIVDSGTTNLRLPKKVFEAAVKSIKAASSTEKFPDGFWLGEQLVCWQAGTTPWNIFPVISLYLMGEVTNQSFRITILPQQYLRPVEDVATSQDDCYKFAISQSSTGTVMGAVIMEGFYVVFDRARKRIGFAVSACHVHDEFRTAAVEGPFVTLDMEDCGYN
3IQF , Knot 121 283 0.80 38 164 270 
MTVAKAIFIKCGNLGTSMMMDMLLDERADREDVEFRVVGTSVKMDPECVEAAVEMALDIAEDFEPDFIVYGGPNPAAPGPSKAREMLADSEYPAVIIGDAPGLKVKDEMEEQGLGYILVKPDAMLGARREFLDPVEMAIYNADLMKVLAATGVFRVVQEAFDELIEKAKEDEISENDLPKLVIDRNTLLEREEFENPYAMVKAMAALEIAENVADVSVEGCFVEQDKERYVPIVASAHEMMRKAAELADEARELEKSNDAVLRTPHAPDGKVLSKRKFMEDPE

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 \(c\).

\(|P_{f(1HOP_1)}(2) \setminus P_{f(2IRZ_1)}(2)|=65\), \(|P_{f(2IRZ_1)}(2) \setminus P_{f(1HOP_1)}(2)|=81\). 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 1:10011111001100101011011000100110001100110011101000110111011100010011101111011111001001000111100011100101111000111001000101101110010111110000110011011011000011001001100001011000010110000110001111011001110100110010001011110110101101001001010000001111101011110010011111010000111111100110001011000100111001000001110011100110100101101001011011001010110011010010001111001011011000111100001110000111011100100100101111011001100000111001101
Pair \(Z_2\) Length of longest common subsequence
1HOP_1,2IRZ_1 146 3
1HOP_1,3IQF_1 154 4
2IRZ_1,3IQF_1 162 4

Newick tree

 
[
	3IQF_1:80.93,
	[
		1HOP_1:73,2IRZ_1:73
	]:7.93
]

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 \((0.85)(0.8)(\frac{836 }{\log_{20} 836}-\frac{405}{\log_{20}405})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 1HOP_1 2IRZ_1 149 144.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]