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Parikh vectors
7RUP_1 3VAK_1 1GKL_1 Letter Amino acid
5 12 19 F Phenylalanine
7 9 18 P Proline
3 5 25 Y Tyrosine
2 9 15 V Valine
3 11 12 D Aspartic acid
2 9 14 I Isoleucine
7 18 23 G Glycine
1 2 13 H Histidine
4 20 16 L Leucine
4 9 12 K Lycine
6 5 9 M Methionine
1 7 15 T Threonine
3 0 4 W Tryptophan
2 10 22 N Asparagine
1 1 3 C Cysteine
6 12 9 Q Glutamine
6 6 16 E Glutamic acid
3 7 20 S Serine
5 15 24 A Alanine
2 7 8 R Arginine 

7RUP_1|Chain A|GRB10-interacting GYF protein 2|Homo sapiens (9606)
>3VAK_1|Chains A, B|Splicing factor U2AF 65 kDa subunit|Homo sapiens (9606)
>1GKL_1|Chains A, B|ENDO-1,4-BETA-XYLANASE Y|CLOSTRIDIUM THERMOCELLUM (1515)
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 \(w=f(c)\)
7RUP , Knot 43 73 0.84 40 65 71 
GPLEMHEAMQKWYYKDPQGEIQGPFNNQEMAEWFQAGYFTMSLLVKRACDESFQPLGDIMKMWGRVPFSPGPA
3VAK , Knot 85 174 0.84 38 135 169 
GPLGSARRLYVGNIPFGITEEAMMDFFNAQMRLGGLTQAPGNPVLAVQINQDKNFAFLEFRSVDETTQAMAFDGIIFQGQSLKIRRPHDYQPLPGAHKLFIGGLPNYLNDDQVKELLTSFGPLKAFNLVKDSATGLSKGYAFCEYVDINVTDQAIAGLNGMQLGDKKLLVQRAS
1GKL , Knot 130 297 0.83 40 199 288 
MASDKFPVAENPSSSFKYESAVQYRPAPDSYLNPCPQAGRIVKETYTGINGTKSLNVYLPYGYDPNKKYNIFYLMHGGGENENTIFSNDVKLQNILDHAIMNGELEPLIVVTPTFNGGNCTAQNFYQEFRQNVIPFVESKYSTYAESTTPQGIAASRMHRGFGGFAMGGLTTWYVMVNCLDYVAYFMPLSGDYWYGNSPQDKANSIAEAINRSGLSKREYFVFAATGSEDIAYANMNPQIEAMKALPHFDYTSDFSKGNFYFLVAPGATHWWGYVRHYIYDALPYFFHELEHHHHHH

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 \(c\).

\(|P_{f(7RUP_1)}(2) \setminus P_{f(3VAK_1)}(2)|=37\), \(|P_{f(3VAK_1)}(2) \setminus P_{f(7RUP_1)}(2)|=107\). 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 1:1110100110010000101010111000011011011010101110010000101110110111011101111
Pair \(Z_2\) Length of longest common subsequence
7RUP_1,3VAK_1 144 3
7RUP_1,1GKL_1 192 3
3VAK_1,1GKL_1 172 4

Newick tree

 
[
	1GKL_1:96.67,
	[
		7RUP_1:72,3VAK_1:72
	]:24.67
]

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 \((0.85)(0.8)(\frac{247 }{\log_{20} 247}-\frac{73}{\log_{20}73})=56.6\)
Status Protein1 Protein2 d d1/2
Query variables 7RUP_1 3VAK_1 72 50.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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