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Parikh vectors
5VPM_1 2JTW_1 8SNI_1 Letter Amino acid
12 0 6 N Asparagine
6 0 4 C Cysteine
22 0 14 I Isoleucine
13 0 15 P Proline
3 1 5 W Tryptophan
17 1 12 Y Tyrosine
16 4 25 A Alanine
10 1 5 R Arginine
12 1 8 Q Glutamine
15 4 23 K Lycine
8 0 11 M Methionine
19 0 15 F Phenylalanine
17 0 13 D Aspartic acid
6 2 16 H Histidine
29 1 15 T Threonine
16 0 22 E Glutamic acid
34 0 25 G Glycine
26 5 28 L Leucine
33 5 16 S Serine
23 0 15 V Valine 

5VPM_1|Chains A, B|Renin|Homo sapiens (9606)
>2JTW_1|Chain A|transmembrane helix 7 of yeast VATPase|null (559292)
>8SNI_1|Chains A, B|(S)-hydroxynitrile lyase|Hevea brasiliensis (3981)
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 \(w=f(c)\)
5VPM , Knot 146 337 0.84 40 205 322 
GNTTSSVILTNYMDTQYYGEIGIGTPPQTFKVVFDTGSSNVWVPSSKCSRLYTACVYHKLFDASDSSSYKHNGTELTLRYSTGTVSGFLSQDIITVGGITVTQMFGEVTEMPALPFMLAEFDGVVGMGFIEQAIGRVTPIFDNIISQGVLKEDVFSFYYNRDSENSQSLGGQIVLGGSDPQHYEGNFHYINLIKTGVWQIQMKGVSVGSSTLLCEDGCLALVDTGASYISGSTSSIEKLMEALGAKKRLFDYVVKCNEGPTLPDISFHLGGKEYTLTSADYVFQESYSSKKLCTLAIHAMDIPPPTGPTWALGATFIRKFYTEFDRRNNRIGFALAR
2JTW , Knot 15 25 0.64 20 22 23 
KKSHTASYLRLWALSLAHAQLSSKK
8SNI , Knot 127 293 0.82 40 180 271 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSMAFAHFVLVHGACHGAWSWHKLKPLLEALGHKVTALDLAASGVDPRQIEEIGSFDEYSEPLLTFLEALPPGEKVILVGHSLGGMNIAIAADKYCEKIAAAVFLAAFLPDTEHCPSYVLDKYNEVTPAENWLDTTFFTYTKDGKEITGMFFGPKFLAHKLYQLCGPEEYELAKMLVRKSSLFMNILAKRPFFTKEGYGSIKKIYVWTDQDKGIPPEFQLWQIENYKPDKVYKVEGGDHMAMLTKTKEIAEILQEVADTYN

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 \(c\).

\(|P_{f(5VPM_1)}(2) \setminus P_{f(2JTW_1)}(2)|=191\), \(|P_{f(2JTW_1)}(2) \setminus P_{f(5VPM_1)}(2)|=8\). 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 1:1000001110001000001011110110010111001000111100000010010100011010000000001001010000101011100011011110100111010011111111101011111111001110101110011001110001101000000000001110111110010000101001011001110101011011000110001011110011001010000100110111100011001100001101101010111000010010011000000001001110110111101101111101100100010000001111110
Pair \(Z_2\) Length of longest common subsequence
5VPM_1,2JTW_1 199 4
5VPM_1,8SNI_1 167 4
2JTW_1,8SNI_1 176 3

Newick tree

 
[
	2JTW_1:97.15,
	[
		5VPM_1:83.5,8SNI_1:83.5
	]:13.65
]

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 \((0.85)(0.8)(\frac{362 }{\log_{20} 362}-\frac{25}{\log_{20}25})=109.\)
Status Protein1 Protein2 d d1/2
Query variables 5VPM_1 2JTW_1 138 73.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]