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Parikh vectors
7UIA_1 4DYE_1 5JXY_1 Letter Amino acid
6 12 17 K Lycine
10 23 9 T Threonine
7 48 8 A Alanine
6 23 11 E Glutamic acid
2 13 7 M Methionine
11 43 12 V Valine
3 11 11 F Phenylalanine
5 10 9 N Asparagine
8 28 10 D Aspartic acid
2 6 4 C Cysteine
6 4 8 Q Glutamine
1 16 8 H Histidine
4 16 13 I Isoleucine
9 9 8 Y Tyrosine
5 21 8 R Arginine
11 38 18 G Glycine
10 34 19 L Leucine
5 20 11 P Proline
14 17 12 S Serine
4 6 1 W Tryptophan 

7UIA_1|Chains A[auth B], C[auth E]|VHH-G6|Vicugna pacos (30538)
>4DYE_1|Chain A|isomerase|Streptomyces coelicolor (1902)
>5JXY_1|Chain A|G/T mismatch-specific thymine DNA glycosylase|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 \(w=f(c)\)
7UIA , Knot 64 129 0.80 40 102 126 
QLQLVETGGGLVKPGGSLRLSCVVSGFTFDDYRMAWVRQAPGKELEWVSSIDSWSINTYYEDSVKGRFTISTDNAKNTLYLQMSSLKPEDTAVYYCAAEDRLGVPTINAHPSKYDYNYWGQGTQVTVSS
4DYE , Knot 163 398 0.81 40 202 372 
MHHHHHHSSGVDLGTENLYFQSMMKITDVDVWVVNLPLVNPFTSSFETKTGETRTVVRVRTDSGVEGWGETMWGAPVAAIVRRMAPDLIGTSPFALEAFHRKQHMVPFFYGYLGYAAIAAVDVACWDAMGKATGQSVTDLLGGAVRDEVPITALITRADAPGATPADLPKAMAEHAVRVVEEGGFDAVKLKGTTDCAGDVAILRAVREALPGVNLRVDPNAAWSVPDSVRAGIALEELDLEYLEDPCVGIEGMAQVKAKVRIPLCTNMCVVRFEDFAPAMRLNAVDVIHGDVYKWGGIAATKALAAHCETFGLGMNLHSGGELGIATAAHLAVVSSTPVLSRAIDSMYYLHADDIIEPLHLENGRLRVPSGPGLGVSVDEDKLRHYAGVNERDGDLTG
5JXY , Knot 97 204 0.84 40 145 198 
GSHMASFNGVSEAELLTKTLPDILTFNLDIVIIGINPGLMAAYKGHHYPGPGNHFWKCLFMSGLSEVQLNHMDDHTLPGKYGIGFTNMVERTTPGSKDLSSKEFREGGRILVQKLQKYQPRIAVFNGKCIYEIFSKEVFGVKVKNLEFGLQPHKIPDTETLCYVMPSSSARCAQFPRAQDKVHYYIKLKDLRDQLKGIERNMDV

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 \(c\).

\(|P_{f(7UIA_1)}(2) \setminus P_{f(4DYE_1)}(2)|=39\), \(|P_{f(4DYE_1)}(2) \setminus P_{f(7UIA_1)}(2)|=139\). 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 1:010110011111011101010011011010000111100111001011001001010000000101010100001000101010010100011000110001111010101000000011010010100
Pair \(Z_2\) Length of longest common subsequence
7UIA_1,4DYE_1 178 3
7UIA_1,5JXY_1 163 3
4DYE_1,5JXY_1 183 3

Newick tree

 
[
	4DYE_1:92.99,
	[
		7UIA_1:81.5,5JXY_1:81.5
	]:11.49
]

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 \((0.85)(0.8)(\frac{527 }{\log_{20} 527}-\frac{129}{\log_{20}129})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 7UIA_1 4DYE_1 146 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]