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Parikh vectors
4XFE_1 7QDQ_1 4AVQ_1 Letter Amino acid
27 9 9 V Valine
0 1 4 C Cysteine
15 5 3 Q Glutamine
20 4 18 I Isoleucine
16 3 13 F Phenylalanine
33 2 12 A Alanine
14 3 7 N Asparagine
25 4 10 G Glycine
15 4 5 P Proline
15 5 12 S Serine
3 0 2 W Tryptophan
11 4 15 R Arginine
22 5 25 E Glutamic acid
16 3 9 M Methionine
25 9 14 K Lycine
16 5 12 T Threonine
8 7 7 Y Tyrosine
18 4 11 D Aspartic acid
16 2 6 H Histidine
26 13 10 L Leucine 

4XFE_1|Chain A|TRAP dicarboxylate transporter subunit DctP|Pseudomonas putida ND6 (231023)
>7QDQ_1|Chain A|E3 ubiquitin-protein ligase Mdm2|Homo sapiens (9606)
>4AVQ_1|Chains A, B, C, D|POLYMERASE PA|INFLUENZA A VIRUS (11320)
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 \(w=f(c)\)
4XFE , Knot 143 341 0.81 38 200 323 
MTFKRKLLLAVLPFAFSVAMPASALDIKFAEIHPAGYPTVVAEQNMGKKLEQASNGDITFKMFAGGVLGSEKEVIEQAQIGAVQMTRVSLGIVGPVVPDVNVFNMPFVFRDHDHMRKIIDGEIGQEILDKITNSDFNLVALAWMDGGSRSIYTKKPVRSLEDLKGMKIRVQGNPLFIDMMNAMGGNGIAMDTGEIFSALQTGVIDGAENNPPTLLEHNHFQSAKYYTLTGHLILPEPVVMSKTTWNKLTPEQQVLVKKVAREAQMEERALWDAKSAASEEKLKAAGVEFITVDKKPFYDATASVREKYGAQYADLMKRIDAVQAENLYFQGHHHHHHHHHH
7QDQ , Knot 47 92 0.77 38 77 88 
PASEQETLVRPKPLLLKLLKSVGAQKDTYTMKEVLFYLGQYIMTKRLYDEKQQHIVYCSNDLLGDLFGVPSFSVKEHRKIYTMIYRNLVVVN
4AVQ , Knot 97 204 0.84 40 145 197 
MGSGMAMEDFVRQCFNPMIVELAEKAMKEYGEDPKIETNKFAAICTHLEVCFMYSDFHFIDERGESIIVESGDPNALLKHRFEIIEGRDRIMAWTVVNSICNTTGVEKPKFLPDLYDYKENRFIEIGVTRREVHIYYLEKANKIKSEKTHIHIFSFTGEEMATKADYTLDEESRARIKTRLFTIRQEMASRSLWDSFRQSERGE

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 \(c\).

\(|P_{f(4XFE_1)}(2) \setminus P_{f(7QDQ_1)}(2)|=149\), \(|P_{f(7QDQ_1)}(2) \setminus P_{f(4XFE_1)}(2)|=26\). 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 1:10100011111111110111110110101101011101011100011001001001010101111111100001100101111010010111111111010110111110000010011010110011001000010111111101100010000110010010110101010111101101111011110010110110011101100011011000010010000101011110111100001001010001110011001010001110100110000101111011010001100101010000110010110010110100101010000000000
Pair \(Z_2\) Length of longest common subsequence
4XFE_1,7QDQ_1 175 3
4XFE_1,4AVQ_1 159 3
7QDQ_1,4AVQ_1 148 3

Newick tree

 
[
	4XFE_1:86.55,
	[
		4AVQ_1:74,7QDQ_1:74
	]:12.55
]

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 \((0.85)(0.8)(\frac{433 }{\log_{20} 433}-\frac{92}{\log_{20}92})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 4XFE_1 7QDQ_1 125 79
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]