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Parikh vectors
3RBI_1 6PAE_1 3CLJ_1 Letter Amino acid
13 8 4 Q Glutamine
18 32 7 G Glycine
3 9 2 M Methionine
13 25 6 T Threonine
13 19 7 R Arginine
1 0 3 C Cysteine
0 0 1 W Tryptophan
18 35 2 V Valine
19 31 11 A Alanine
9 14 7 N Asparagine
10 21 12 D Aspartic acid
16 14 7 E Glutamic acid
17 28 17 L Leucine
10 18 20 S Serine
14 6 10 H Histidine
18 20 17 I Isoleucine
14 16 12 K Lycine
4 6 5 F Phenylalanine
9 12 2 P Proline
11 13 5 Y Tyrosine 

3RBI_1|Chains A, B|Sortase family protein|Streptococcus agalactiae serogroup V (216466)
>6PAE_1|Chains A, B, C, D|L-asparaginase|Dickeya chrysanthemi (556)
>3CLJ_1|Chain A|Protein NRD1|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 \(w=f(c)\)
3RBI , Knot 106 230 0.83 38 153 224 
MRGSHHHHHHGSSHANINAFKEAVTKIDRVEINRRLELAYAYNASIAGAKTNGEYPALKDPYSAEQKQAGVVEYARMLEVKEQIGHVIIPRINQDIPIYAGSAEENLQRGVGHLEGTSLPVGGESTHAVLTAHRGLPTAKLFTNLDKVTVGDRFYIEHIGGKIAYQVDQIKVIAPDQLEDLYVIQGEDHVTLLTCTPYMINSHRLLVRGKRIPYVEKTVQKDSKTFRQQQ
6PAE , Knot 138 327 0.81 36 184 310 
ADKLPNIVILATGGTIAGSAATGTQTTGYKAGALGVDTLINAVPEVKKLANVKGEQFSNMASENMTGDVVLKLSQRVNELLARDDVDGVVITHGTDTVEESAYFLHLTVKSDKPVVFVAAMRPATAISADGPMNLLEAVRVAGDKQSRGRGVMVVINDRIGSARYITKTNASTLDTFRANEEGYLGVIIGNRIYYQNRIDKLHTTRSVFDVRGLTSLPKVDILYGYQDDPEYLYDAAIQHGVKGIVYAGMGAGSVSVRGIAGMRKALEKGVVVMRSTRTGNGIVPPDEELPGLVSDSLNPAHARILLMLALTRTSDPKVIQEYFHTY
3CLJ , Knot 76 157 0.81 40 112 150 
MDFQNFVATLESFKDLKSGISGSRIKKLTTYALDHIDIESKIISLIIDYSRLCPDSHKLGSLYIIDSIGRAYLDETRSNSNSSSNKPGTCAHAINTLGEVIQELLSDAIAKSNQDHKEKIRMLLDIWDRSGLFQKSYLNAIRSKCFAAALEHHHHHH

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 \(c\).

\(|P_{f(3RBI_1)}(2) \setminus P_{f(6PAE_1)}(2)|=62\), \(|P_{f(6PAE_1)}(2) \setminus P_{f(3RBI_1)}(2)|=93\). 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 1:10100000001000101011001100100101000101101001011110001001110010010000111100101101000110111101000111011010001001110101001111100001110100111010110010010110010100111011001001011110010010110100010110001011000011101001101000100000010000
Pair \(Z_2\) Length of longest common subsequence
3RBI_1,6PAE_1 155 4
3RBI_1,3CLJ_1 141 6
6PAE_1,3CLJ_1 162 4

Newick tree

 
[
	6PAE_1:81.98,
	[
		3RBI_1:70.5,3CLJ_1:70.5
	]:11.48
]

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 \((0.85)(0.8)(\frac{557 }{\log_{20} 557}-\frac{230}{\log_{20}230})=93.3\)
Status Protein1 Protein2 d d1/2
Query variables 3RBI_1 6PAE_1 115 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]