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Parikh vectors
8TGX_1 2PJO_1 6UCT_1 Letter Amino acid
4 17 23 T Threonine
8 15 13 V Valine
6 17 15 G Glycine
7 23 29 I Isoleucine
7 14 14 F Phenylalanine
5 19 23 S Serine
1 1 0 W Tryptophan
8 14 17 Y Tyrosine
5 24 7 A Alanine
0 2 2 C Cysteine
10 22 31 L Leucine
6 15 10 P Proline
4 17 19 N Asparagine
9 8 17 D Aspartic acid
12 15 29 E Glutamic acid
3 4 13 H Histidine
14 2 21 K Lycine
3 4 4 M Methionine
8 21 16 R Arginine
5 14 16 Q Glutamine 

8TGX_1|Chain A|Protein lgg-1|Caenorhabditis elegans (6239)
>2PJO_1|Chain A|Ricin (EC 3.2.2.22)|Ricinus communis (3988)
>6UCT_1|Chain A|p9-1|Mal de Rio Cuarto virus (185954)
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 \(w=f(c)\)
8TGX , Knot 62 125 0.79 38 95 122 
GSMKWAYKEENNFEKRRAEGDKIRRKYPDRIPVIVEKAPKSKLHDLDKKKYLVPSDLTVGQFYFLIRKRIQLRPEDALFFFVNNVIPQTMTTMGQLYQDHHEEDLFLYIAYSDESVYGGEVEKKE
2PJO , Knot 116 268 0.80 40 171 260 
MIFPKQYPIINFTTAGATVQSYTNFIRAVRGRLTTGADVRHEIPVLPNRVGLPINQRFILVELSNHAELSVTLALDVTNAYVVGYRAGNSAYFFHPDNQEDAEAITHLFTDVQNRYTFAFGGNYDRLEQLAGNLRENIELGNGPLEEAISALYYYSTGGTQLPTLARSFIICIQMISEAARFQYIEGEMRTRIRYNRRSAPDPSVITLENSWGRLSTAIQESNQGAFASPIQLQRRNGSKFSVYDVSILIPIIALMVYRCAPPPSSQF
6UCT , Knot 137 319 0.82 38 190 300 
MHHHHHHADLERRTFGSYKIEEITIKNDQQQKTTNQQQISNNEQRISTKKIPILDDGIFDLINYLLNGTHFDKTHYCGFDYSHLPTLERDFNTASNYVSENYSIIVEEIDLNKYERSESISLKSPDFTVVLEYFKKHVEGQTEQEENKTESTSSELPAKIVRELPLLPIMCRESEDSISEDILEGEGAVIQVLKMFMKGFLVHLGENPNSYDRQLTIEKYRPLLISIIGYEFTVGTRATHTKINHIYYQLATFDNYPFDLLRFQLQSLIDTPNVIKERIEKDGLFKVITTTNARGQYQSVLLRGINGSESYLNLKRYRK

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 \(c\).

\(|P_{f(8TGX_1)}(2) \setminus P_{f(2PJO_1)}(2)|=46\), \(|P_{f(2PJO_1)}(2) \setminus P_{f(8TGX_1)}(2)|=122\). 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 1:10101100000010000101001000010011111001100010010000011100101101011100010101001111110011100100110100000000111011000001011010000
Pair \(Z_2\) Length of longest common subsequence
8TGX_1,2PJO_1 168 3
8TGX_1,6UCT_1 181 3
2PJO_1,6UCT_1 157 4

Newick tree

 
[
	8TGX_1:90.05,
	[
		2PJO_1:78.5,6UCT_1:78.5
	]:11.55
]

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 \((0.85)(0.8)(\frac{393 }{\log_{20} 393}-\frac{125}{\log_{20}125})=81.2\)
Status Protein1 Protein2 d d1/2
Query variables 8TGX_1 2PJO_1 103 75
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]