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Parikh vectors
4ACR_1 2JJX_1 1DXU_1 Letter Amino acid
26 7 1 Q Glutamine
32 12 4 E Glutamic acid
14 14 10 H Histidine
27 13 11 S Serine
6 1 1 W Tryptophan
13 7 3 Y Tyrosine
20 18 4 N Asparagine
18 11 11 K Lycine
10 8 2 M Methionine
15 8 7 F Phenylalanine
25 8 7 P Proline
14 1 1 C Cysteine
58 23 18 L Leucine
20 10 9 T Threonine
26 23 13 V Valine
43 18 21 A Alanine
38 15 3 R Arginine
28 15 8 D Aspartic acid
32 22 7 G Glycine
13 21 0 I Isoleucine 

4ACR_1|Chains A, B, C, D|GLYPICAN-1|HOMO SAPIENS (9606)
>2JJX_1|Chains A, B, C|URIDYLATE KINASE|BACILLUS ANTHRACIS (198094)
>1DXU_1|Chains A, C|HEMOGLOBIN (DEOXY) (ALPHA CHAIN)|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 \(w=f(c)\)
4ACR , Knot 194 478 0.83 40 238 439 
APQLHHHHHHDLYENLYFQGKLDPASKSRSCGEVRQIYGAKGFSLSDVPQAEISGEHLRICPQGYTCCTSEMEENLANRSHAELETALRDSSRVLQAMLATQLRSFDDHFQHLLNDSERTLQATFPGAFGELYTQNARAFRDLYSELRLYYRGANLHLEETLAEFWARLLERLFKQLHPQLLLPDDYLDCLGKQAEALRPFGEAPRELRLRATRAFVAARSFVQGLGVASDVVRKVAQVPLGPECSRAVMKLVYCAHCLGVPGARPCPDYCRNVLKGCLANQADLDAEWRNLLDSMVLITDKFWGTSGVESVIGSVHTWLAEAINALQDNRDTLTAKVIQGCGNPKVNPQGPGPEEKRRRGKLAPRERPPSGTLEKLVSEAKAQLRDVQDFWISLPGTLCSEKMALSTASDDRCWNGMARGRYLPEVMGDGLANQINNPEVEVDITKPDMTIRQQIMQLKIMTNRLRSAYNGNDVDFQ
2JJX , Knot 112 255 0.81 40 165 245 
MAHHHHHHMRPYKRVLIKLSGGALADQTGNSFNSKRLEHIANEILSIVDLGIEVSIVIGGGNIFRGHLAEEWGIDRVEADNIGTLGTIINSLMLRGVLTSKTNKEVRVMTSIPFNAVAEPYIRLRAVHHLDNGYIVIFGGGNGQPFVTTDYPSVQRAIEMNSDAILVAKQGVDGVFTSDPKHNKSAKMYRKLNYNDVVRQNIQVMDQAALLLARDYNLPAHVFNFDEPGVMRRICLGEHVGTLINDDASLLVHEK
1DXU , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR

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 \(c\).

\(|P_{f(4ACR_1)}(2) \setminus P_{f(2JJX_1)}(2)|=123\), \(|P_{f(2JJX_1)}(2) \setminus P_{f(4ACR_1)}(2)|=50\). 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 1:1101000000010001010101011000000101001011011010011010101001010101000000010001100001010011000001101111001001000100110000001010111111010000101100100010100011010100011011101100110010101111000100110010110111011001010100111110011011111001100110111110000111011001001111110101000001101011001010101001100111100011100110011101001110110110000001010110101010101011110000001011100011010100110010101001001110111010000111001000001011101001101110111001001010101001010100011010110001001001001010
Pair \(Z_2\) Length of longest common subsequence
4ACR_1,2JJX_1 173 6
4ACR_1,1DXU_1 195 4
2JJX_1,1DXU_1 154 5

Newick tree

 
[
	4ACR_1:96.69,
	[
		2JJX_1:77,1DXU_1:77
	]:19.69
]

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 \((0.85)(0.8)(\frac{733 }{\log_{20} 733}-\frac{255}{\log_{20}255})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 4ACR_1 2JJX_1 169 128
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]