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Parikh vectors
3JSO_1 3VRC_1 6NEL_1 Letter Amino acid
5 1 19 H Histidine
22 8 13 L Leucine
1 4 7 Y Tyrosine
19 8 11 V Valine
6 9 7 N Asparagine
0 2 4 C Cysteine
13 3 18 I Isoleucine
6 5 13 T Threonine
1 1 3 W Tryptophan
17 32 14 A Alanine
13 5 5 Q Glutamine
5 3 8 M Methionine
9 3 7 P Proline
16 8 29 E Glutamic acid
17 12 12 G Glycine
10 9 14 K Lycine
6 3 15 F Phenylalanine
9 3 13 S Serine
15 5 18 R Arginine
12 7 11 D Aspartic acid 

3JSO_1|Chains A, B|LexA repressor|Escherichia coli K-12 (83333)
>3VRC_1|Chains A, B|Cytochrome c'|Thermochromatium tepidum (1050)
>6NEL_1|Chain A|Polymerase acidic protein|Influenza A virus (655278)
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 \(w=f(c)\)
3JSO , Knot 95 202 0.83 38 137 187 
MKALTARQQEVFDLIRDHISQTGMPPTRAEIAQRLGFRSPNAAEEHLKALARKGVIEIVSGASRGIRLLQEEEEGLPLVGRVAAGEPLLAQQHIEGHYQVDPSLFKPNADFLLRVSGMSMKDIGIMDGDLLAVHKTQDVRNGQVVVARIDDEVTVARLKKQGNKVELLPENSEFKPIVVDLRQQSFTIEGLAVGVIRNGDWL
3VRC , Knot 63 131 0.78 40 89 120 
ADLSPEEQIETRQAGYAFMAWNMGKIKANLEGEYNADQVRAAANVVAAIANSGMGALYGPGTDKNVGAVKTRAKPELFQNLEDVGKLARDLGTAANALAAAAATGEANAVKSAFADVGAACKACHQKYRAD
6NEL , Knot 110 241 0.83 40 162 224 
MAHHHHHHSRAWRHPQFGGHHHHHHALEVLFQGPLGSMEDFVRQCFNPMIVELAEKAMKEYGEDPKIETNKFAAICTHLEVCFMYSDFHFIDERGESIIVESGDPNALLKHRFEIIEGRDRIMAWTVVNSICNTTGVEKPKFLPDLYDYKENRFIEIGVTRREVHIYYLEKANKIKSEKTHIHIFSFTGEEMATKADYTLDEESRARIKTRLFTIRQEMASRSLWDSFRQSERGEETVEER

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 \(c\).

\(|P_{f(3JSO_1)}(2) \setminus P_{f(3VRC_1)}(2)|=99\), \(|P_{f(3VRC_1)}(2) \setminus P_{f(3JSO_1)}(2)|=51\). 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 1:1011010000110110001000111100101100111001011000101110011101101100110110000011111101111011110001010001010110101011101011010011110101111000001001011110100010110100010010111000010111101000010101111111001011
Pair \(Z_2\) Length of longest common subsequence
3JSO_1,3VRC_1 150 3
3JSO_1,6NEL_1 155 3
3VRC_1,6NEL_1 163 3

Newick tree

 
[
	6NEL_1:80.97,
	[
		3JSO_1:75,3VRC_1:75
	]:5.97
]

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 \((0.85)(0.8)(\frac{333 }{\log_{20} 333}-\frac{131}{\log_{20}131})=62.0\)
Status Protein1 Protein2 d d1/2
Query variables 3JSO_1 3VRC_1 79 62.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]