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Parikh vectors
4HPP_1 7YOE_1 6UDP_1 Letter Amino acid
9 11 14 N Asparagine
11 2 8 C Cysteine
18 6 22 Q Glutamine
22 20 34 S Serine
6 7 14 M Methionine
23 14 37 D Aspartic acid
28 22 27 E Glutamic acid
14 4 11 H Histidine
7 26 37 I Isoleucine
24 27 38 V Valine
35 14 22 R Arginine
66 25 45 L Leucine
5 21 37 K Lycine
8 6 19 F Phenylalanine
6 2 0 W Tryptophan
11 7 17 Y Tyrosine
61 34 47 A Alanine
34 33 47 G Glycine
36 15 16 P Proline
19 20 27 T Threonine 

4HPP_1|Chain A|Probable glutamine synthetase|Pseudomonas aeruginosa (208964)
>7YOE_1|Chain A|Cysteine synthase|Haemophilus influenzae Rd KW20 (71421)
>6UDP_1|Chains A, B, C, D, E, F, G, H|Inosine-5'-monophosphate dehydrogenase 2|Homo sapiens (9606)
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 \(w=f(c)\)
4HPP , Knot 174 443 0.79 40 202 408 
MNRLQPVRLVSFVTTDLAGITRGRSLPLATLEEQLASGCGWVPANSSLTPQDLIDESSPWGSHGDLRLLPDPNSRVRVEQGPDAAAPALDYLHGNLVETDGTPWPACPRSLLRAEVERYRDSGLQVIAAFEHEFSLLGLPGERPAAAFSLQAQRAAGQFPGWLVSALAQAGTEPEMFLPEYGQRQYEVTCRPAQGVAAADRAVNVREVTREVARQMGLRTCFAPLPAPGAVTNGVHLHLSLQHADGSPLLYEPGRPNDLSELGEHWAAGVLAHLPALCALTAPTAASYLRLKPHHWSAAYACLGLRNREAALRICPVVSVGGKPLGKQYNLEFRPMDATTCPHLAMAAVLIAGRLGIERRLPLRALADVDPHGLSDEERQARGIQALPATLGDALDCLQRDEALCAELPKPLLDTYLAMKRHELALTAGLSDDDLCRHYAELY
7YOE , Knot 134 316 0.81 40 183 298 
MAIYADNSYSIGNTPLVRLKHFGHNGNVVVKIEGRNPSYSVKCRIGANMVWQAEKDGTLTKGKEIVDPTSGNTGIALAYVAAARGYKITLTMPETMSLERKRLLCGLGVNLVLTEGAKGMKGAIAKAEEIVASDPSRYVMLKQFENPANPQIHRETTGPEIWKDTDGKVDVVVAGVGTGGSITGISRAIKLDFGKQITSVAVEPVESPVISQTLAGEEVKPGPHKIQGIGAGFIPKNLDLSIIDRVETVDSDTALATARRLMAEEGILAGISSGAAVAAADRLAKLPEFADKLIVVILPSASERYLSTALFEGIEG
6UDP , Knot 207 519 0.83 38 243 482 
SEFELMADYLISGGTSYVPDDGLTAQQLFNCGDGLTYNDFLILPGYIDFTADQVDLTSALTKKITLKTPLVSSPMDTVTEAGMAIAMALTGGIGFIHHNCTPEFQANEVRKVKKYEQGFITDPVVLSPKDRVRDVFEAKARHGFCGIPITDTGRMGSRLVGIISSRDIDFLKEEEHDCFLEEIMTKREDLVVAPAGITLKEANEILQRSKKGKLPIVNEDDELVAIIARTDLKKNRDYPLASKDAKKQLLCGAAIGTHEDDKYRLDLLAQAGVDVVVLDSSQGNSIFQINMIKYIKDKYPNLQVIGGNVVTAAQAKNLIDAGVDALRVGMGSGSICITQEVLACGRPQATAVYKVSEYARRFGVPVIADGGIQNVGHIAKALALGASTVMMGSLLAATTEAPGEYFFSDGIRLKKYRGMGSLDAMDKHLSSQNRYFSEADKIKVAQGVSGAVQDKGSIHKFVPYLIAGIQHSCQDIGAKSLTQVRAMMYSGELKFEKRTSSAQVEGGVHSLHSYEKRLF

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 \(c\).

\(|P_{f(4HPP_1)}(2) \setminus P_{f(7YOE_1)}(2)|=87\), \(|P_{f(7YOE_1)}(2) \setminus P_{f(4HPP_1)}(2)|=68\). 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 1:10010110110110001111001001111010001101011111000101001100001110010101110100010100110111111001010110001011110100110101000000110111110001011111100111110101001110111111011101100101111001000001000110111110011010010001100111000111111111100110101010010101110011010010011001111111011110110110110010101001011010111000011101011101110111000010101101000101111111111011100011101110101011000000101101111011011001000011010110111000111000011101110000100001010
Pair \(Z_2\) Length of longest common subsequence
4HPP_1,7YOE_1 155 6
4HPP_1,6UDP_1 149 5
7YOE_1,6UDP_1 136 5

Newick tree

 
[
	4HPP_1:78.50,
	[
		6UDP_1:68,7YOE_1:68
	]:10.50
]

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 \((0.85)(0.8)(\frac{759 }{\log_{20} 759}-\frac{316}{\log_{20}316})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 4HPP_1 7YOE_1 146 127.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]