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Parikh vectors
3TLE_1 4GWN_1 8QKY_1 Letter Amino acid
19 16 4 H Histidine
36 41 13 L Leucine
4 12 2 C Cysteine
6 28 2 Q Glutamine
33 34 11 I Isoleucine
27 58 10 S Serine
18 20 10 A Alanine
17 32 6 N Asparagine
20 24 9 E Glutamic acid
26 20 8 K Lycine
24 35 8 T Threonine
13 24 4 Y Tyrosine
4 11 1 W Tryptophan
15 29 8 V Valine
14 30 8 R Arginine
22 38 8 D Aspartic acid
30 41 16 G Glycine
7 11 7 M Methionine
19 29 6 F Phenylalanine
17 20 7 P Proline 

3TLE_1|Chains A, B|MccF|Escherichia coli (562)
>4GWN_1|Chain A|Meprin A subunit beta|Homo sapiens (9606)
>8QKY_1|Chains A, B, C|Deoxyuridine 5'-triphosphate nucleotidohydrolase|Escherichia phage T5 (2695836)
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 \(w=f(c)\)
3TLE , Knot 154 371 0.81 40 211 343 
MGHHHHHHHHHHHHSSGHIDDDDKHMLEMIQSHPLLAAPLAVGDTIGFFSSSAPATVTAKNRFFRGVEFLQRKGFKLVSGKLTGKTDFYRSGTIKERAQEFNELVYNPDITCIMSTIGGDNSNSLLPFLDYDAIIANPKIIIGYSDTTALLAGIYAKTGLITFYGPALIPSFGEHPPLVDITYESFIKILTRKQSGIYTYTLPEKWSDESINWNENKILRPKKLYKNNCAFYGSGKVEGRVIGGNLNTLTGIWGSEWMPEIRNGDILFIEDSRKSIATVERLFSMLKLNRVFDKVSAIILGKHELFDCAGSKRRPYEVLTEVLDGKQIPVLDGFDCSHTHPMLTLPLGVKLAIDFDNKNISITEQYLSTEK
4GWN , Knot 227 553 0.86 40 279 521 
NSIIGEKYRWPHTIPYVLEDSLEMNAKGVILNAFERYRLKTCIDFKPWAGETNYISVFKGSGCWSSVGNRRVGKQELSIGANCDRIATVQHEFLHALGFWHEQSRSDRDDYVRIMWDRILSGREHNFNTYSDDISDSLNVPYDYTSVMHYSKTAFQNGTEPTIVTRISDFEDVIGQRMDFSDSDLLKLNQLYNCSSSLSFMDSCSFELENVCGMIQSSGDNADWQRVSQVPRGPESDHSNMGQCQGSGFFMHFDSSSVNVGATAVLESRTLYPKRGFQCLQFYLYNSGSESDQLNIYIREYSADNVDGNLTLVEEIKEIPTGSWQLYHVTLKVTKKFRVVFEGRKGSGASLGGLSIDDINLSETRCPHHIWHIRNFTQFIGSPNGTLYSPPFYSSKGYAFQIYLNLAHVTNAGIYFHLISGANDDQLQWPCPWQQATMTLLDQNPDIRQRMSNQRSITTDPFMTTDNGNYFWDRPSKVGTVALFSNGTQFRRGGGYGTSAFITHERLKSRDFIKGDDVYILLTVEDISHLNSTQIQLTPAPSVQDLCSKTTCK
8QKY , Knot 70 148 0.78 40 113 141 
MIKIKLTHPDCMPKIGSEDAAGMDLRAFFGTNPAADLRAIAPGKSLMIDTGVAVEIPRGWFGLVVPRSSLGKRHLMIANTAGVIDSDYRGTIKMNLYNYGSEMQTLENFERLCQLVVLPHYSTHNFKIVDELEETIRGEGGFGSSGSK

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 \(c\).

\(|P_{f(3TLE_1)}(2) \setminus P_{f(4GWN_1)}(2)|=42\), \(|P_{f(4GWN_1)}(2) \setminus P_{f(3TLE_1)}(2)|=110\). 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 1:11000000000000001010000001101100011111111110011110001110101000110110110001101101010100010001010001001001100101001100111000001111100011110101111000001111110100111010111111011001111010000110110000011000011001000010100001101001000001101010101011110100101111001110100101111000000110100110110100110010111110001100110000100110011010011110110000001110111110111010000101000010000
Pair \(Z_2\) Length of longest common subsequence
3TLE_1,4GWN_1 152 4
3TLE_1,8QKY_1 162 4
4GWN_1,8QKY_1 210 3

Newick tree

 
[
	8QKY_1:98.98,
	[
		3TLE_1:76,4GWN_1:76
	]:22.98
]

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 \((0.85)(0.8)(\frac{924 }{\log_{20} 924}-\frac{371}{\log_{20}371})=147.\)
Status Protein1 Protein2 d d1/2
Query variables 3TLE_1 4GWN_1 189 154.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]