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Parikh vectors
1LJI_1 2VSU_1 3VYD_1 Letter Amino acid
2 10 8 M Methionine
6 7 3 W Tryptophan
3 8 17 Y Tyrosine
12 24 34 G Glycine
3 8 19 F Phenylalanine
2 11 13 P Proline
10 9 33 S Serine
6 13 22 I Isoleucine
8 29 28 L Leucine
7 14 30 T Threonine
11 17 10 R Arginine
14 10 12 N Asparagine
2 25 16 E Glutamic acid
1 2 6 H Histidine
6 18 15 K Lycine
6 16 23 V Valine
12 26 16 A Alanine
7 13 17 D Aspartic acid
8 4 6 C Cysteine
3 12 12 Q Glutamine 

1LJI_1|Chains A, B|Lysozyme C|Gallus gallus (9031)
>2VSU_1|Chains A, B, D|P-HYDROXYCINNAMOYL COA HYDRATASE/LYASE|PSEUDOMONAS FLUORESCENS (294)
>3VYD_1|Chains A, B|Renin|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 \(w=f(c)\)
1LJI , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
2VSU , Knot 129 276 0.87 40 186 269 
MSTYEGRWKTVKVEIEDGIAFVILNRPEKRNAMSPTLNREMIDVLETLEQDPAAGVLVLTGAGEAWTAGMDLKEYFREVDAGPEILQEKIRREASQWQWKLLRMYAKPTIAMVNGWCFGGGFAPLVACDLAICADEATFGLSEINWGIPPGNLVSKAMADTVGHRQSLYYIMTGKTFGGQKAAEMGLVNESVPLAQLREVTIELARNLLEKNPVVLRAAKHGFKRCRELTWEQNEDYLYAKLDQSRLLDTEGGREQGMKQFLDDKSIKPGLQAYKR
3VYD , Knot 145 340 0.82 40 205 324 
LTLGNTTSSVILTNYMDTQYYGEIGIGTPPQTFKVVFDTGSSNVWVPSSKCSRLYTACVYHKLFDASDSSSYKHNGTELTLRYSTGTVSGFLSQDIITVGGITVTQMFGEVTEMPALPFMLAEFDGVVGMGFIEQAIGRVTPIFDNIISQGVLKEDVFSFYYNRDSENSQSLGGQIVLGGSDPQHYEGNFHYINLIKTGVWQIQMKGVSVGSSTLLCEDGCLALVDTGASYISGSTSSIEKLMEALGAKKRLFDYVVKCNEGPTLPDISFHLGGKEYTLTSADYVFQESYSSKKLCTLAIHAMDIPPPTGPTWALGATFIRKFYTEFDRRNNRIGFALAR

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 \(c\).

\(|P_{f(1LJI_1)}(2) \setminus P_{f(2VSU_1)}(2)|=51\), \(|P_{f(2VSU_1)}(2) \setminus P_{f(1LJI_1)}(2)|=133\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
1LJI_1,2VSU_1 184 3
1LJI_1,3VYD_1 171 3
2VSU_1,3VYD_1 173 3

Newick tree

 
[
	2VSU_1:90.52,
	[
		1LJI_1:85.5,3VYD_1:85.5
	]:5.02
]

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 \((0.85)(0.8)(\frac{405 }{\log_{20} 405}-\frac{129}{\log_{20}129})=83.3\)
Status Protein1 Protein2 d d1/2
Query variables 1LJI_1 2VSU_1 110 80
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]