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Parikh vectors
6MJZ_1 3JPW_1 3ETV_1 Letter Amino acid
26 17 15 A Alanine
28 28 19 D Aspartic acid
31 25 22 E Glutamic acid
57 34 24 I Isoleucine
34 20 21 T Threonine
17 16 10 Y Tyrosine
18 12 10 R Arginine
32 11 33 N Asparagine
23 14 17 Q Glutamine
44 29 44 L Leucine
4 13 10 M Methionine
12 15 17 F Phenylalanine
36 32 35 S Serine
14 3 3 C Cysteine
29 20 17 G Glycine
17 16 4 P Proline
6 7 5 H Histidine
33 19 27 K Lycine
2 7 4 W Tryptophan
32 25 18 V Valine 

6MJZ_1|Chains A, B, C|Fusion glycoprotein F0|Human parainfluenza virus 3 (11216)
>3JPW_1|Chain A|Glutamate [NMDA] receptor subunit epsilon-2|Rattus norvegicus (10116)
>3ETV_1|Chain A|Protein transport protein TIP20, Protein transport protein DSL1 chimera|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
6MJZ , Knot 199 495 0.83 40 229 469 
QIDITKLQHVGVLVNSPKGMKISQNFETRYLILSLIPKIEDSNSCGDQQIKQYKRLLDRLIIPLYDGLKLQKDVIVTNQESNENTDPRTERFFGGVIGTIALGVATSAQITAAVALVEAKQAKSDIEKLKEAIRDTNKAVQSVCSSVGNCIVAIKSVQDYVNKEIVPSIARLGCEAAGLQLGIALTQHYSELTNCFGDNIGSLQEKGIKLQCIASLYRTNITEIFTTSTVDKYDIYDLLFTESIKVRVIDVDLNDYSITLQVRLPLLTRLLNTQIYKVDSISYNIQNREWYIPLPSHIMTKGAFLGGADVKECIEAFSSYICPSDPGFVLNHEMESCLSGNISQCPRTTVTSDIVPRYAFVNGGVVANCITTTCTCNGIGNRINQPPDQGVKIITHKECNTIGINGMLFNTNKEGTLAFYTPDDITLNNSVALDPIDISIELNKVKSDLEESKEWYRRSNQKLSAIEDKIEEILSKIYHIENEIARIKKLIGEAP
3JPW , Knot 155 363 0.84 40 223 345 
PPSIGIAVILVGTSDEVAIKDAHEKDDFHHLSVVPRVELVAMNETDPKSIITRICDLMSDRKIQGVVFADDTDQEAIAQILDFISAQTLTPILGIHGGSSMIMADKDESSMFFQFGPSIEQQASVMLNIMEEYDWYIFSIVTTYFPGYQDFVNKIRSTIENSFVGWELEEVLLLDMSLDDGDSKIQNQLKKLQSPIILLYCTKEEATYIFEVANSVGLTGYGYTWIVPSLVAGDTDTVPSEFPTGLISVSYDEWDYGLPARVRDGIAIITTAASDMLSEHSFIPEPKSSCYNTHEKRIYQSNMLNRYLINVTFEGRDLSFSEDGYQMHPKLVIILLNKERKWERVGKWKDKSLQMKYYVWPRM
3ETV , Knot 149 355 0.82 40 206 326 
GAMGMNGIDDLLNINDRIKQVQNERNELASKLQNLKQSLASNDTGGGSGGGSDSSDLLQREAILANELNILDNLKTFLNLIKEVKTNLNILELENCYYSLQSLRKKMRNNAAYLKQSFNFQQSISTYVDTLHLELVSTLYKILTNGFWKITENSIQFTPTVEWGKDKVHIEYDTFMDFVAQQYFPKGSLDNQAWFILDMTSADSQEQVRAKLNTIMKEYMNLSRIVSMIKNSIFISGKEISYENEKNILVFSKSSSHGQHCVSTVLTSFEAVCDFMLDGLAFRDRKTLSYELGPLFNTEFTKFVKNNASIILESLDSPLKNLVSVINNKLTRLVAKSEVTNWTHSGKEIQDLLMN

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 \(c\).

\(|P_{f(6MJZ_1)}(2) \setminus P_{f(3JPW_1)}(2)|=75\), \(|P_{f(3JPW_1)}(2) \setminus P_{f(6MJZ_1)}(2)|=69\). 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 1:010100100111110010110100010000111011101000000100010000011001111100110100011100000000001000011111110111111001010111111010010001001001100000110010001100111100100010001110110110011110111110000001000110011010001101001101000010011000010000100111000101011010100001010101111001100010010010001000010111100110011111110100010110001010011111000100010101000100010001110011101111100100000001110010011001101100000001110111100000101110010010100011101101010100100010000010000000101100010011001001000110100111011
Pair \(Z_2\) Length of longest common subsequence
6MJZ_1,3JPW_1 144 4
6MJZ_1,3ETV_1 149 4
3JPW_1,3ETV_1 157 3

Newick tree

 
[
	3ETV_1:77.97,
	[
		6MJZ_1:72,3JPW_1:72
	]: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 \((0.85)(0.8)(\frac{858 }{\log_{20} 858}-\frac{363}{\log_{20}363})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 6MJZ_1 3JPW_1 165 144
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]