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Parikh vectors
3TRJ_1 4OWG_1 2AUX_1 Letter Amino acid
15 24 24 G Glycine
6 4 2 H Histidine
5 2 4 M Methionine
15 12 16 S Serine
3 4 13 Y Tyrosine
23 27 18 A Alanine
4 5 8 C Cysteine
13 17 10 E Glutamic acid
18 15 8 I Isoleucine
20 15 12 L Leucine
8 21 19 K Lycine
7 8 5 F Phenylalanine
14 9 19 N Asparagine
12 12 9 D Aspartic acid
8 10 9 Q Glutamine
0 5 4 W Tryptophan
12 25 15 V Valine
2 8 7 R Arginine
6 10 8 P Proline
10 15 5 T Threonine 

3TRJ_1|Chains A, B, C, D|Phosphoheptose isomerase|Francisella tularensis subsp. tularensis (119856)
>4OWG_1|Chains A, B|Triosephosphate isomerase|Oryctolagus cuniculus (9986)
>2AUX_1|Chain A|Cathepsin K|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 \(w=f(c)\)
3TRJ , Knot 93 201 0.81 38 134 190 
SNAMTSLDKINSYFESSIQAKIETANALPPAIAQAAKAMVSCLENGGKVLVCGNGSSGVIAQHFTSKLLNHFEMERPPLPAIALTGDVATITAVGNHYGFSQIFAKQVAALGNEDDILLVITTSGDSENILSAVEEAHDLEMKVIALTGGSGGALQNMYNTDDIELRVPSDNIANIQENHFLIVHCLCDIIDQKLFAGLED
4OWG , Knot 113 248 0.83 40 178 237 
APSRKFFVGGNWKMNGRKKNLGELITTLNAAKVPADTEVVCAPPTAYIDFARQKLDPKIAVAAQNCYKVTNGAFTGEISPGMIKDCGATWVVLGHSERRHVFGESDELIGQKVAHALSEGLGVIACIGEKLDEREAGITEKVVFEQTKVIADNVKDWSKVVLAYEPVWAIGTGKTATPQQAQEVHEKLRGWLKSNVSDAVAQSTRIIYGGSVTGATCKELASQPDVDGFLVGGASLKPEFVDIINAKQ
2AUX , Knot 99 215 0.82 40 148 209 
APDSVDYRKKGYVTPVKNQGQCGSCWAFSSVGALEGQLKKKTGKLLNLSPQNLVDCVSENDGCGGGYMTNAFQYVQKNRGIDSEDAYPYVGQEESCMYNPTGKAAKCRGYREIPEGNEKALKRAVARVGPVSVAIDASLTSFQFYSKGVYYDESCNSDNLNHAVLAVGYGIQKGNKHWIIKNSWGENWGNKGYILMARNKNNACGIANLASFPKM

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 \(c\).

\(|P_{f(3TRJ_1)}(2) \setminus P_{f(4OWG_1)}(2)|=62\), \(|P_{f(4OWG_1)}(2) \setminus P_{f(3TRJ_1)}(2)|=106\). 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 1:001100100100010001010100101111111011011100100110111010100111100100011001010011111111010110101110001100111001111100001111100010000110110010010101111011011110010000010101100011010000111100100110001111100
Pair \(Z_2\) Length of longest common subsequence
3TRJ_1,4OWG_1 168 4
3TRJ_1,2AUX_1 168 4
4OWG_1,2AUX_1 168 3

Newick tree

 
[
	2AUX_1:84,
	[
		3TRJ_1:84,4OWG_1:84
	]:0
]

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 \((0.85)(0.8)(\frac{449 }{\log_{20} 449}-\frac{201}{\log_{20}201})=72.5\)
Status Protein1 Protein2 d d1/2
Query variables 3TRJ_1 4OWG_1 96 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]