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Parikh vectors
2LRP_1 3IBH_1 8FXY_1 Letter Amino acid
5 13 7 P Proline
6 3 1 W Tryptophan
13 14 16 I Isoleucine
15 10 15 D Aspartic acid
9 18 13 E Glutamic acid
20 17 11 G Glycine
7 7 5 F Phenylalanine
9 12 17 T Threonine
4 13 7 R Arginine
8 7 14 N Asparagine
0 2 9 C Cysteine
2 7 4 Q Glutamine
9 24 13 L Leucine
12 18 11 K Lycine
6 9 8 M Methionine
11 15 11 V Valine
7 18 7 A Alanine
20 10 13 S Serine
7 10 15 Y Tyrosine
2 6 7 H Histidine 

2LRP_1|Chain A|Endoglucanase H|Clostridium thermocellum (203119)
>3IBH_1|Chain A|Saccharomyces cerevisiae Gtt2|Saccharomyces cerevisiae (4932)
>8FXY_1|Chains A, C[auth B], E[auth D], G[auth F], I, K|CPXV040 protein|Cowpox virus (Brighton Red) (265872)
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 \(w=f(c)\)
2LRP , Knot 82 172 0.81 38 122 167 
MASAVGEKMLDDFEGVLNWGSYSGEGAKVSTKIVSGKTGNGMEVSYTGTTDGYWGTVYSLPDGDWSKWLKISFDIKSVDGSANEIRFMIAEKSINGVGDGEHWVYSITPDSSWKTIEIPFSSFRRRLDYQPPGQDMSGTLDLDNIDSIHFMYANNKSGKFVVDNIKLIGALE
3IBH , Knot 110 233 0.85 40 161 228 
MNGRGFLIYNGGEKMKQKMIIYDTPAGPYPARVRIALAEKNMLSSVQFVRINLWKGEHKKPEFLAKNYSGTVPVLELDDGTLIAECTAITEYIDALDGTPTLTGKTPLEKGVIHMMNKRAELELLDPVSVYFHHATPGLGPEVELYQNKEWGLRQRDKALHGMHYFDTVLRERPYVAGDSFSMADITVIAGLIFAAIVKLQVPEECEALRAWYKRMQQRPSVKKLLEIRSKSS
8FXY , Knot 97 204 0.84 40 146 199 
KPEAEYKNTICPPRQDYRYWYFAAELTIGVNYDINSTIMGECHMSESYIDRNANIVLTGYGLEINMTIMDTDQRFVAAAEGVGKDNKLSVLLFTTQRLDKVHHNISVTITCMEMNCGTTKYDSDLPESIHHKSSCDITINGSCVTCVNLETDPTKINPHYLHPKDKYLYRNSEYGMRGSYGVTFMDELNQCFLDIKEVSYDICY

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 \(c\).

\(|P_{f(2LRP_1)}(2) \setminus P_{f(3IBH_1)}(2)|=66\), \(|P_{f(3IBH_1)}(2) \setminus P_{f(2LRP_1)}(2)|=105\). 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 1:1101110011001011101100010110100011010010110100010001011010011010100110101010010101001011110001011101001100101000100101110010001000111001010101001001011010000101110010111110
Pair \(Z_2\) Length of longest common subsequence
2LRP_1,3IBH_1 171 4
2LRP_1,8FXY_1 152 4
3IBH_1,8FXY_1 177 3

Newick tree

 
[
	3IBH_1:90.38,
	[
		2LRP_1:76,8FXY_1:76
	]:14.38
]

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 \((0.85)(0.8)(\frac{405 }{\log_{20} 405}-\frac{172}{\log_{20}172})=69.3\)
Status Protein1 Protein2 d d1/2
Query variables 2LRP_1 3IBH_1 89 75.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]