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Parikh vectors
8VCE_1 3QMB_1 6KRE_1 Letter Amino acid
4 8 8 C Cysteine
18 2 44 I Isoleucine
9 3 13 M Methionine
14 11 27 R Arginine
27 3 37 D Aspartic acid
10 7 9 H Histidine
16 6 41 K Lycine
23 2 15 P Proline
10 1 30 T Threonine
5 0 0 W Tryptophan
10 3 8 Y Tyrosine
17 3 52 A Alanine
32 3 58 L Leucine
11 4 12 F Phenylalanine
29 0 48 V Valine
15 2 24 N Asparagine
6 5 16 Q Glutamine
17 5 36 E Glutamic acid
23 5 36 G Glycine
31 6 45 S Serine 

8VCE_1|Chains A, B|Probable carboxylesterase 120|Arabidopsis thaliana (3702)
>3QMB_1|Chain A|CpG-binding protein|Homo sapiens (9606)
>6KRE_1|Chains A, B[auth a]|T-complex protein 1 subunit alpha|Saccharomyces cerevisiae S288c (559292)
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 \(w=f(c)\)
8VCE , Knot 143 327 0.84 40 195 310 
MSEPSPIADPYAYLNIVNNPDGSITRDLSNFPCTAATPDPSPLNPAVSKDLPVNQLKSTWLRLYLPSSAVNEGNVSSQKLPIVVYYHGGGFILCSVDMQLFHDFCSEVARDLNAIVVSPSYRLAPEHRLPAAYDDGVEALDWIKTSDDEWIKSHADFSNVFLMGTSAGGNLAYNVGLRSVDSVSDLSPLQIRGLILHHPFFGGEERSESEIRLMNDQVCPPIVTDVMWDLSLPVGVDRDHEYSNPTVGDGSEKLEKIGRLRWKVMMIGGEDDPMIDLQKDVAKLMKKKGVEVVEHYTGGHVHGAEIRDPSKRKTLFLSIKNFIFSVL
3QMB , Knot 42 79 0.77 36 65 74 
MHHHHHHSSRENLYFQGQIKRSARMCGECEACRRTEDCGHCDFCRDMKKFGGPNKIRQKCRLRQCQLRARESYKYFPSS
6KRE , Knot 220 559 0.83 38 233 516 
MSQLFNNSRSDTLFLGGEKISGDDIRNQNVLATMAVANVVKSSLGPVGLDKMLVDDIGDFTVTNDGATILSLLDVQHPAGKILVELAQQQDREIGDGTTSVVIIASELLKRANELVKNKIHPTTIITGFRVALREAIRFINEVLSTSVDTLGKETLINIAKTSMSSKIIGADSDFFSNMVVDALLAVKTQNSKGEIKYPVKAVNVLKAHGKSATESLLVPGYALNCTVASQAMPKRIAGGNVKIACLDLNLQKARMAMGVQINIDDPEQLEQIRKREAGIVLERVKKIIDAGAQVVLTTKGIDDLCLKEFVEAKIMGVRRCKKEDLRRIARATGATLVSSMSNLEGEETFESSYLGLCDEVVQAKFSDDECILIKGTSKHSSSSIILRGANDYSLDEMERSLHDSLSVVKRTLESGNVVPGGGCVEAALNIYLDNFATTVGSREQLAIAEFAAALLIIPKTLAVNAAKDSSELVAKLRSYHAASQMAKPEDVKRRSYRNYGLDLIRGKIVDEIHAGVLEPTISKVKSLKSALEACVAILRIDTMITVDPEPPKEDPHDH

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 \(c\).

\(|P_{f(8VCE_1)}(2) \setminus P_{f(3QMB_1)}(2)|=163\), \(|P_{f(3QMB_1)}(2) \setminus P_{f(8VCE_1)}(2)|=33\). 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 1:100101110101010110010101000100110011010101101110001110010001101011001100101000011111000111111001010110010001100101111010001110001111000110110110000001100010100111110011101100111001001001011010111100111110000000101100010111100111010111110000000010110100010011010101111110001110100011011000110110000110101101001000001110100111011
Pair \(Z_2\) Length of longest common subsequence
8VCE_1,3QMB_1 196 3
8VCE_1,6KRE_1 162 4
3QMB_1,6KRE_1 226 3

Newick tree

 
[
	3QMB_1:11.81,
	[
		8VCE_1:81,6KRE_1:81
	]:31.81
]

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 \((0.85)(0.8)(\frac{406 }{\log_{20} 406}-\frac{79}{\log_{20}79})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 8VCE_1 3QMB_1 128 79
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]