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Parikh vectors
4CEC_1 3JVA_1 5HYH_1 Letter Amino acid
5 9 10 N Asparagine
19 32 20 G Glycine
4 8 7 M Methionine
13 8 14 S Serine
4 1 8 W Tryptophan
15 22 26 V Valine
7 20 19 D Aspartic acid
14 5 20 H Histidine
3 9 11 F Phenylalanine
11 25 19 T Threonine
2 6 2 C Cysteine
11 30 27 E Glutamic acid
9 33 27 L Leucine
12 27 5 K Lycine
6 9 12 P Proline
4 6 7 Y Tyrosine
14 44 29 A Alanine
5 7 25 R Arginine
10 15 13 Q Glutamine
15 38 16 I Isoleucine 

4CEC_1|Chains A, B|INTEGRASE|HUMAN IMMUNODEFICIENCY VIRUS 1 (11676)
>3JVA_1|Chains A, B, C, D, E, F, G, H|Dipeptide Epimerase|Enterococcus faecalis V583 (226185)
>5HYH_1|Chain A|Uncharacterized protein|Streptomyces venezuelae (953739)
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 \(w=f(c)\)
4CEC , Knot 82 183 0.77 40 136 177 
MGSSHHHHHHSSGLVPRGSHMHGQVDSSPGIWQLDCTHLEGKVILVAVHVASGYIEAEVIPAETGQETAYFLLKLAGRWPVKTVHTDNGSNFTSTTVKAACWWAGIKQEDGIPYNPQSQGVIESMNKELKKIIGQVRDQAEHLKTAVQMAVFIHNHKRKGGIGGYSAGERIVDIIATDIQTKE
3JVA , Knot 146 354 0.80 40 172 328 
MKIKQVHVRASKIKLKETFTIALGTIESADSAIVEIETEEGLVGYGEGGPGIFITGETLAGTLETIELFGQAIIGLNPFNIEKIHEVMDKISAFAPAAKAAIDIACYDLMGQKAQLPLYQLLGGYDNQVITDITLGIDEPNVMAQKAVEKVKLGFDTLKIKVGTGIEADIARVKAIREAVGFDIKLRLDANQAWTPKDAVKAIQALADYQIELVEQPVKRRDLEGLKYVTSQVNTTIMADESCFDAQDALELVKKGTVDVINIKLMKCGGIHEALKINQICETAGIECMIGCMAEETTIGITAAAHLAAAQKNITRADLDATFGLETAPVTGGVSLEAKPLLELGEAAGLGISH
5HYH , Knot 137 317 0.83 40 194 304 
HHHHHHHHAIAENAVINRLVGNWHRRAAVKREEPDVYALFDPGRPDFREDMIPFRGHPIWERLSDETRSRLLSWGWVAYNRNTVLIEQRIANPAFELVIGGAYPGLGGQQLELAVAQAMVDEQYHTLMHINGSAVTRRMRRSDFSDRVLPDSHITTIHQEHLDRCEEPWQRSLTTLGFATVAEISINAYLDLLADDQEIQVVNSTTVKLHNRDEYCHASISGEMMKQVYEALPADRRRFLLEKVVAGLEAFVAPDFTTWESIVAFEGVPGWEKAAAEVREAQGGTHLVQDHSGIHTLLTEMDVLDQVEFGWGTTVTR

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 \(c\).

\(|P_{f(4CEC_1)}(2) \setminus P_{f(3JVA_1)}(2)|=62\), \(|P_{f(3JVA_1)}(2) \setminus P_{f(4CEC_1)}(2)|=98\). 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 1:110000000000111101001010100011110100001010111111011010101011110010001011101110111001000010010000101101111100001110010001110010001001110100010010011011111000000111110011001101110010000
Pair \(Z_2\) Length of longest common subsequence
4CEC_1,3JVA_1 160 3
4CEC_1,5HYH_1 170 6
3JVA_1,5HYH_1 166 3

Newick tree

 
[
	5HYH_1:85.29,
	[
		4CEC_1:80,3JVA_1:80
	]:5.29
]

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 \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{183}{\log_{20}183})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 4CEC_1 3JVA_1 125 94.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]