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Parikh vectors
1YIW_1 2ZNH_1 2FPZ_1 Letter Amino acid
2 26 4 F Phenylalanine
2 31 14 A Alanine
5 23 15 D Aspartic acid
1 5 12 H Histidine
7 35 9 K Lycine
7 23 11 T Threonine
1 10 10 Y Tyrosine
4 28 25 V Valine
0 4 9 W Tryptophan
4 22 12 R Arginine
0 6 8 C Cysteine
7 22 11 I Isoleucine
0 12 4 M Methionine
10 40 20 L Leucine
3 21 23 P Proline
3 32 10 S Serine
2 23 5 N Asparagine
6 13 13 Q Glutamine
6 38 9 E Glutamic acid
6 18 21 G Glycine 

1YIW_1|Chains A, B, C|Ubiquitin|null
>2ZNH_1|Chains A, B|Antithrombin-III|Homo sapiens (9606)
>2FPZ_1|Chains A, B, C, D|Tryptase beta-2|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 \(w=f(c)\)
1YIW , Knot 40 76 0.76 34 61 74 
LQIFVKTLTGKTITLEVEPSDTIENVKAKIQDKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLRGG
2ZNH , Knot 181 432 0.84 40 224 400 
HGSPVDICTAKPRDIPMNPMCIYRSPEKKATEDEGSEQKIPEATNRRVWELSKANSRFATTFYQHLADSKNDNDNIFLSPLSISTAFAMTKLGACNDTLQQLMEVFKFDTISEKTSDQIHFFFAKLNCRLYRKANKSSKLVSANRLFGDKSLTFNETYQDISELVYGAKLQPLDFKENAEQSRAAINKWVSNKTEGRITDVIPSEAINELTVLVLVNTIYFKGLWKSKFSPENTRKELFYKADGESCSASMMYQEGKFRYRRVAEGTQVLELPFKGDDITMVLILPKPEKSLAKVEKELTPEVLQEWLDELEEMMLVVHMPRFRIEDGFSLKEQLQDMGLVDLFSPEKSKLPGIVAEGRDDLYVSDAFHKAFLEVNEEGSEAAASTAVVIAGRSLNPNRVTFKANRPFLVFIREVPLNTIIFMGRVANPCVK
2FPZ , Knot 116 245 0.86 40 180 239 
IVGGQEAPRSKWPWQVSLRVHGPYWMHFCGGSLIHPQWVLTAAHCVGPDVKDLAALRVQLREQHLYYQDQLLPVSRIIVHPQFYTAQIGADIALLELEEPVKVSSHVHTVTLPPASETFPPGMPCWVTGWGDVDNDERLPPPFPLKQVKVPIMENHICDAKYHLGAYTGDDVRIVRDDMLCAGNTRRDSCQGDSGGPLVCKVNGTWLQAGVVSWGEGCAQPNRPGIYTRVTYYLDWIHHYVPKKP

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 \(c\).

\(|P_{f(1YIW_1)}(2) \setminus P_{f(2ZNH_1)}(2)|=16\), \(|P_{f(2ZNH_1)}(2) \setminus P_{f(1YIW_1)}(2)|=179\). 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 1:1011100101001010101000100101010000111100001111100100100100001000001011101011
Pair \(Z_2\) Length of longest common subsequence
1YIW_1,2ZNH_1 195 3
1YIW_1,2FPZ_1 189 2
2ZNH_1,2FPZ_1 188 4

Newick tree

 
[
	1YIW_1:96.67,
	[
		2FPZ_1:94,2ZNH_1:94
	]:2.67
]

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 \((0.85)(0.8)(\frac{508 }{\log_{20} 508}-\frac{76}{\log_{20}76})=130.\)
Status Protein1 Protein2 d d1/2
Query variables 1YIW_1 2ZNH_1 165 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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