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Parikh vectors
6HFW_1 4PSW_1 8PRT_1 Letter Amino acid
43 32 4 L Leucine
15 15 6 N Asparagine
12 9 2 Q Glutamine
19 26 2 E Glutamic acid
24 20 6 I Isoleucine
34 11 4 T Threonine
31 21 4 V Valine
2 2 0 C Cysteine
11 2 1 M Methionine
22 21 2 F Phenylalanine
27 9 4 P Proline
46 15 4 A Alanine
17 27 6 K Lycine
28 26 9 S Serine
8 6 2 W Tryptophan
13 16 6 Y Tyrosine
30 15 0 R Arginine
29 28 4 D Aspartic acid
48 10 13 G Glycine
16 6 5 H Histidine 

6HFW_1|Chains A, B|Decaprenylphosphoryl-beta-D-ribose oxidase|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>4PSW_1|Chain A|Histone acetyltransferase type B catalytic subunit|Saccharomyces cerevisiae (559292)
>8PRT_1|Chain A|Cell surface protein|Dictyoglomus thermophilum (309799)
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 \(w=f(c)\)
6HFW , Knot 191 475 0.82 40 234 446 
MGSSHHHHHHSQDPMLSVGATTTATRLTGWGRTAPSVANVLRTPDAEMIVKAVARVAESGGGRGAIARGLGRSYGDNAQNGGGLVIDMTPLNTIHSIDADTKLVDIDAGVNLDQLMKAALPFGLWVPVLPGTRQVTVGGAIACDIHGKNHHSAGSFGNHVRSMDLLTADGEIRHLTPTGEDAELFWATVGGNGLTGIIMRATIEMTPTSTAYFIADGDVTASLDETIALHSDGSEARYTYSSAWFDAISAPPKLGRAAVSRGRLATVEQLPAKLRSEPLKFDAPQLLTLPDVFPNGLANKYTFGPIGELWYRKSGTYRGKVQNLTQFYHPLDMFGEWNRAYGPAGFLQYQFVIPTEAVDEFKKIIGVIQASGHYSFLNVFKLFGPRNQAPLSFPIPGWNICVDFPIKDGLGKFVSELDRRVLEFGGRLYTAKDSRTTAETFHAMYPRVDEWISVRRKVDPLRVFASDMARRLELL
4PSW , Knot 138 317 0.83 40 190 306 
NDFKPETWTSSANEALRVSIVGENAVQFSPLFTYPIYGDSEKIYGYKDLIIHLAFDSVTFKPYVNVKYSAKLGDDNIVDVEKKLLSFLPKDDVIVRDEAKWVDCFAEERKTHNLSDVFEKVSEYSLNGEEFVVYKSSLVDDFARRMHRRVQIFSLLFIEAANYIDETDPSWQIYWLLNKKTKELIGFVTTYKYWHYLGAKSFDEDIDKKFRAKISQFLIFPPYQNKGHGSCLYEAIIQSWLEDKSITEITVEDPNEAFDDLRDRNDIQRLRKLGYDAVFQKHSDLSDEFLESSRKSLKLEERQFNRLVEMLLLLNNS
8PRT , Knot 28 84 0.49 36 41 44 
GSHMTGGSVHSSPAIGQDGTIYVGSNDHYLYAINPNGKLKWKFETGGSVHSSPAIGQDGTIYVGSNDHYLYAINPNGKLKWKFE

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 \(c\).

\(|P_{f(6HFW_1)}(2) \setminus P_{f(4PSW_1)}(2)|=107\), \(|P_{f(4PSW_1)}(2) \setminus P_{f(6HFW_1)}(2)|=63\). 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 1:1100000000000111011100010010111001101101100101011101110110011101111011100010010011111101011001001010001101011101001101111111111111100010111111001010000011011001001011010101001010100101111011101101111010101010001011101010101000111000100100000011101101110110111001011010011101000110101101101101110111000011111011000010001010010010011011101001011111100011110011001001111101010001101101111000111011111101010111001110110010001101110100100000010010110101001101000101101110011001011
Pair \(Z_2\) Length of longest common subsequence
6HFW_1,4PSW_1 170 3
6HFW_1,8PRT_1 225 3
4PSW_1,8PRT_1 187 3

Newick tree

 
[
	8PRT_1:10.89,
	[
		6HFW_1:85,4PSW_1:85
	]:23.89
]

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 \((0.85)(0.8)(\frac{792 }{\log_{20} 792}-\frac{317}{\log_{20}317})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 6HFW_1 4PSW_1 168 139
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]