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Parikh vectors
1VHM_1 4ZDG_1 7JRI_1 Letter Amino acid
15 11 40 G Glycine
6 5 6 K Lycine
9 21 22 S Serine
2 3 8 W Tryptophan
14 11 18 D Aspartic acid
10 4 17 Q Glutamine
10 10 12 F Phenylalanine
5 9 5 Y Tyrosine
21 14 74 L Leucine
3 7 6 M Methionine
11 26 21 T Threonine
17 8 40 V Valine
17 15 67 A Alanine
3 5 4 C Cysteine
8 9 14 H Histidine
13 14 9 I Isoleucine
9 7 40 R Arginine
8 23 5 N Asparagine
10 6 39 E Glutamic acid
4 7 35 P Proline 

1VHM_1|Chains A, B|Protein yebR|Escherichia coli (562)
>4ZDG_1|Chains A, B, C|Fiber protein|Human adenovirus 14p1 (814922)
>7JRI_1|Chains A, B|Photoreceptor-histidine kinase BphP|Stigmatella aurantiaca (41)
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 \(w=f(c)\)
1VHM , Knot 92 195 0.83 40 147 187 
MSLLILIGWQIQVRQPSDYIMNKTEFYADLNRDFNALMAGETSFLATLANTSALLYERLTDINWAGFYLLEDDTLVLGPFQGKIACVRIPVGRGVCGTAVARNQVQRIEDVHVFDGHIACDAASNSEIVLPLVVKNQIIGVLDIDSTVFGRFTDEDEQGLRQLVAQLEKVLATTDYKKFFASVAGEGGSHHHHHH
4ZDG , Knot 97 215 0.80 40 151 204 
MRGSHHHHHHGSNSNNICIDDNINTLWTGVNPTEANCQMMDSSESNDCKLILTLVKTGALVTAFVYVIGVSNNFNMLTTYRNINFTAELFFDSAGNLLTSLSSLKTPLNHKSGQNMATGAITNAKSFMPSTTAYPFNNNSRENYIYGTCHYTASDHTAFPIDISVMLNQRAIRADTSYCIRITWSWNTGDAPEGQTSATTLVTSPFTFYYIREDD
7JRI , Knot 187 482 0.80 40 193 421 
DLSQCDREPIHLLGGIQSHGVLLAFRGPDRLLEVVSANAQALLGRPPETLLGQPVGRVLPAEVLAQWEPLVARGSVRVVLPAGAYRALLHESDGLTVLELEPAELQPGMEETALEVVRRLVSPLAGVKGTQALLQTAADTVRALTGFDRVMVYRFDADWHGEVLAESKRGGMDGFLGMHFPATDIPVQARALYTRNPLRLIADARARPVPLLPPVVPALGRPLDLSNSALRSVSPVHLEYLRNMGVGASFSLSLLKEGVLWGLIACHHLEPLHISHERRRACEVLTQLLALQLSAEERAAEASEDAHRAALLGQLATAMGEGGTLEEVLEKESERVLALTGAAGVALLLGEEPLLVGCTPAQDEVEALVAWLATQPFQTSFHTDRLGTVYPPLAARADVAAGILAVRLAPAAARFAIWFRPEVARTISWAGNPRKPAEPEPGHQRLHPRGSFQAWEETVRDTSLPWKRADLGAAEGFRGALV

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 \(c\).

\(|P_{f(1VHM_1)}(2) \setminus P_{f(4ZDG_1)}(2)|=87\), \(|P_{f(4ZDG_1)}(2) \setminus P_{f(1VHM_1)}(2)|=91\). 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 1:101111111010100100011000010101000101111100011101100011100010010111101100001111110101101011110110101110001001001011010110011000011111110001111101000111010000001100111010011100000011101110110000000
Pair \(Z_2\) Length of longest common subsequence
1VHM_1,4ZDG_1 178 8
1VHM_1,7JRI_1 158 4
4ZDG_1,7JRI_1 190 4

Newick tree

 
[
	4ZDG_1:96.00,
	[
		1VHM_1:79,7JRI_1:79
	]:17.00
]

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 \((0.85)(0.8)(\frac{410 }{\log_{20} 410}-\frac{195}{\log_{20}195})=63.4\)
Status Protein1 Protein2 d d1/2
Query variables 1VHM_1 4ZDG_1 80 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]