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Parikh vectors
3TGL_1 6JXS_1 8VUC_1 Letter Amino acid
13 10 12 P Proline
12 6 8 N Asparagine
13 17 5 E Glutamic acid
19 19 22 G Glycine
6 12 5 H Histidine
17 12 4 I Isoleucine
22 14 17 L Leucine
15 12 7 D Aspartic acid
8 9 9 Q Glutamine
25 16 30 S Serine
26 9 18 T Threonine
15 8 15 Y Tyrosine
17 24 16 A Alanine
7 0 5 C Cysteine
7 18 11 K Lycine
1 9 3 M Methionine
22 16 21 V Valine
10 4 6 R Arginine
10 14 6 F Phenylalanine
4 2 4 W Tryptophan 

3TGL_1|Chain A|TRIACYL-GLYCEROL ACYLHYDROLASE|Rhizomucor miehei (4839)
>6JXS_1|Chains A, B, C, D|FMN-dependent NADH-azoreductase|Bacillus smithii (1479)
>8VUC_1|Chains A, C[auth B]|S1CE2 VARIANT OF FAB-EPR-1 heavy 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 \(w=f(c)\)
3TGL , Knot 121 269 0.84 40 179 256 
SINGGIRAATSQEINELTYYTTLSANSYCRTVIPGATWDCIHCDATEDLKIIKTWSTLIYDTNAMVARGDSEKTIYIVFRGSSSIRNWIADLTFVPVSYPPVSGTKVHKGFLDSYGEVQNELVATVLDQFKQYPSYKVAVTGHSLGGATVLLCALDLYQREEGLSSSNLFLYTQGQPRVGDPAFANYVVSTGIPYRRTVNERDIVPHLPPAAFGFLHAGEEYWITDNSPETVQVCTSDLETSDCSNSIVPFTSVLDHLSYFGINTGLCT
6JXS , Knot 105 231 0.82 38 155 223 
MGSSHHHHHHSSGLVPRGSHMAKVLYITAHPLDETQSYSMAVGKAFIDTYKEVNPNDEVIHIDLYKENIPQIDADVFSGWGKLQSGKGFEELTAEEKAKVGRLAELSDQFVAADKYVFVTPMWNFSFPPVMKAYIDSVAVAGKTFKYTEQGSVGLLTDKKALHIQASGGIFSEGPAAEMEMGHRYLQAIMNFFGVPSFEGLFVEGQNAMPDKAQEIKEKAIARAKDLAHTF
8VUC , Knot 99 224 0.79 40 145 216 
EVQLVESGGGLVQPGGSLRLSCAASGFNLRSYYMHWVRQAPGKGLEWVASISPYYSYTYYADSVKGRFTISADTSKNTAYLQMNSLRAEDTAVYYCARHGYGAMDYWGQGTLVTVFNQIQGPSVFPLAPSSKSTSGGTAALGCLVKDYFPGPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSCDKTHT

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 \(c\).

\(|P_{f(3TGL_1)}(2) \setminus P_{f(6JXS_1)}(2)|=97\), \(|P_{f(6JXS_1)}(2) \setminus P_{f(3TGL_1)}(2)|=73\). 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 1:01011101100001001000001010000001111101001000100010110010011000011110100000101110100010011101011110011101001001110001010001110110010001000111010011110111011010000011000011100010101101111001100111000010000111011111111101100011000010010100001000000001111001100100111001100
Pair \(Z_2\) Length of longest common subsequence
3TGL_1,6JXS_1 170 3
3TGL_1,8VUC_1 172 3
6JXS_1,8VUC_1 164 4

Newick tree

 
[
	3TGL_1:86.63,
	[
		6JXS_1:82,8VUC_1:82
	]:4.63
]

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 \((0.85)(0.8)(\frac{500 }{\log_{20} 500}-\frac{231}{\log_{20}231})=77.4\)
Status Protein1 Protein2 d d1/2
Query variables 3TGL_1 6JXS_1 99 89.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]