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Parikh vectors
3WAN_1 1PST_1 1DLM_1 Letter Amino acid
4 7 11 H Histidine
9 31 26 L Leucine
8 13 18 P Proline
12 10 9 S Serine
4 7 16 N Asparagine
10 7 27 D Aspartic acid
10 5 14 Q Glutamine
8 7 19 E Glutamic acid
4 12 13 Y Tyrosine
6 29 29 G Glycine
5 15 17 T Threonine
4 27 28 A Alanine
9 19 16 I Isoleucine
9 6 9 K Lycine
0 14 2 W Tryptophan
11 17 22 V Valine
9 8 17 R Arginine
1 3 1 C Cysteine
4 5 6 M Methionine
9 24 11 F Phenylalanine 

3WAN_1|Chains A, B|Autophagy-related protein 13, Microtubule-associated proteins 1A/1B light chain 3A|Homo sapiens (9606)
>1PST_1|Chain A[auth L]|PHOTOSYNTHETIC REACTION CENTER|Rhodobacter sphaeroides (1063)
>1DLM_1|Chains A, B|CATECHOL 1,2-DIOXYGENASE|Acinetobacter sp. (62977)
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 \(w=f(c)\)
3WAN , Knot 69 136 0.83 38 111 134 
GSSSGNTHDDFVMIGSPSDRPFKQRRSFADRCKEVQQIRDQHPSKIPVIIERYKGEKQLPVLDKTKFLVPDHVNMSELVKIIRRRLQLNPTQAFFLLVNQHSMVSVSTPIADIYEQEKDEDGFLYMVYASQETFGF
1PST , Knot 117 266 0.81 40 165 249 
FERKYRVPGGTLVGGNLFDFWVGPFYVGFFGVATFFFAALGIILIAWSAVLQGTWNPQLISVYPPALEYGLGGAPLAKGGLWQIITICATGAFVSWALREVEICRKLGIGYHIPFAFAFAILAYLTLVLFRPVMMGAWGYAFPYGIWTHLDWVSNTGYTYGNFHYNPAHMIAISFFFTNALALALHGALVLSAANPEKGKEMRTPDHEDTFFRDLVGYSIGTLGIHRLGLLLSLSAVFFSALCMIITGTIWFDQWVDWWQWWVKLP
1DLM , Knot 135 311 0.83 40 186 298 
MEVKIFNTQDVQDFLRVASGLEQEGGNPRVKQIIHRVLSDLYKAIEDLNITSDEYWAGVAYLNQLGANQEAGLLSPGLGFDHYLDMRMDAEDAALGIENATPRTIEGPLYVAGAPESVGYARMDDGSDPNGHTLILHGTIFDADGKPLPNAKVEIWHANTKGFYSHFDPTGEQQAFNMRRSIITDENGQYRVRTILPAGYGCPPEGPTQQLLNQLGRHGNRPAHIHYFVSADGHRKLTTQINVAGDPYTYDDFAYATREGLVVDAVEHTDPEAIKANDVEGPFAEMVFDLKLTRLVDGVDNQVVDRPRLAV

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 \(c\).

\(|P_{f(3WAN_1)}(2) \setminus P_{f(1PST_1)}(2)|=63\), \(|P_{f(1PST_1)}(2) \setminus P_{f(3WAN_1)}(2)|=117\). 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 1:1000100000111110100011000001100000100100001001111100001000111100001111001010011011000101010011111100001101001110100000000111011010000111
Pair \(Z_2\) Length of longest common subsequence
3WAN_1,1PST_1 180 3
3WAN_1,1DLM_1 179 3
1PST_1,1DLM_1 151 5

Newick tree

 
[
	3WAN_1:94.02,
	[
		1DLM_1:75.5,1PST_1:75.5
	]:18.52
]

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 \((0.85)(0.8)(\frac{402 }{\log_{20} 402}-\frac{136}{\log_{20}136})=80.1\)
Status Protein1 Protein2 d d1/2
Query variables 3WAN_1 1PST_1 102 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]