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Parikh vectors
3VPZ_1 1OXX_1 5PHP_1 Letter Amino acid
11 28 23 K Lycine
15 11 20 T Threonine
46 20 25 A Alanine
17 15 20 N Asparagine
14 23 14 D Aspartic acid
14 24 26 E Glutamic acid
4 6 11 M Methionine
6 0 7 C Cysteine
16 8 15 Q Glutamine
28 26 27 G Glycine
34 34 24 L Leucine
12 16 18 P Proline
25 22 18 S Serine
2 2 8 W Tryptophan
7 6 19 Y Tyrosine
22 42 14 V Valine
9 20 18 R Arginine
12 6 18 H Histidine
19 30 20 I Isoleucine
18 14 19 F Phenylalanine 

3VPZ_1|Chain A|Glucokinase|Pseudoalteromonas (1161203)
>1OXX_1|Chain A[auth K]|ABC transporter, ATP binding protein|Sulfolobus solfataricus (2287)
>5PHP_1|Chain A|Lysine-specific demethylase 4D|Homo sapiens (9606)
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 \(w=f(c)\)
3VPZ , Knot 140 331 0.81 40 188 314 
SLHSSAQFDPILVADIGGTNARFALITAFDAAKNEFVIEYNHTFPSADFGSLQNATRHYLSTVPHIKPVRACLAVAGPIKAGQVHLTNLGWHFSVSEFKQAFSFLQLEVINDFAAFAYAAPYLDSNQNVVIKAGQADENSNIAVMGPGTGFGAACLVRTAQSSAVLSSEGGHISLAAVTDLDAKLLIELRKEHPHVSLETVFSGPGIAHLYKAMAAVNGITAKHLDAAQISNLANTGECEVCDATLNQFCDWLGSAAGDLALAYGALGGLFIGGGILPRMQSRLLESRFVERFSQKGIMSQYNGQVPVTLVTQDNIPLIGAAACLHNSKQE
1OXX , Knot 147 353 0.81 38 182 327 
MVRIIVKNVSKVFKKGKVVALDNVNINIENGERFGILGPSGAGKTTFMRIIAGLDVPSTGELYFDDRLVASNGKLIVPPEDRKIGMVFQTWALYPNLTAFENIAFPLTNMKMSKEEIRKRVEEVAKILDIHHVLNHFPRELSGAQQQRVALARALVKDPSLLLLDEPFSNLDARMRDSARALVKEVQSRLGVTLLVVSHDPADIFAIADRVGVLVKGKLVQVGKPEDLYDNPVSIQVASLIGEINELEGKVTNEGVVIGSLRFPVSVSSDRAIIGIRPEDVKLSKDVIKDDSWILVGKGKVKVIGYQGGLFRITITPLDSEEEIFTYSDHPIHSGEEVLVYVRKDKIKVFEKN
5PHP , Knot 160 364 0.86 40 229 347 
MHHHHHHSSGVDLGTENLYFQSMETMKSKANCAQNPNCNIMIFHPTKEEFNDFDKYIAYMESQGAHRAGLAKIIPPKEWKARETYDNISEILIATPLQQVASGRAGVFTQYHKKKKAMTVGEYRHLANSKKYQTPPHQNFEDLERKYWKNRIYNSPIYGADISGSLFDENTKQWNLGHLGTIQDLLEKECGVVIEGVNTPYLYFGMWKTTFAWHTEDMDLYSINYLHLGEPKTWYVVPPEHGQRLERLARELFPGSSRGCGAFLRHKVALISPTVLKENGIPFNRITQEAGEFMVTFPYGYHAGFNHGFNCAEAINFATPRWIDYGKMASQCSCGEARVTFSMDAFVRILQPERYDLWKRGQDR

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 \(c\).

\(|P_{f(3VPZ_1)}(2) \setminus P_{f(1OXX_1)}(2)|=82\), \(|P_{f(1OXX_1)}(2) \setminus P_{f(3VPZ_1)}(2)|=76\). 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 1:0100010101111101110010111101101100011100000110101101001000010011010110101111111011010100111010100100110110101100111110111010000011101101000001111111011111011001000111000110101111001010111010000101010011011111010011111011010010110100110010001001010010011101110111101111111111111101000110001100100011100001011101100001111111101000000
Pair \(Z_2\) Length of longest common subsequence
3VPZ_1,1OXX_1 158 4
3VPZ_1,5PHP_1 183 3
1OXX_1,5PHP_1 161 4

Newick tree

 
[
	5PHP_1:88.43,
	[
		3VPZ_1:79,1OXX_1:79
	]:9.43
]

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 \((0.85)(0.8)(\frac{684 }{\log_{20} 684}-\frac{331}{\log_{20}331})=97.2\)
Status Protein1 Protein2 d d1/2
Query variables 3VPZ_1 1OXX_1 123 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]