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Parikh vectors
8ITM_1 5EWV_1 9ESJ_1 Letter Amino acid
22 5 23 V Valine
12 6 8 N Asparagine
23 5 16 I Isoleucine
28 7 19 A Alanine
57 14 39 L Leucine
10 1 3 C Cysteine
5 8 21 K Lycine
21 4 15 S Serine
16 5 17 T Threonine
10 0 4 W Tryptophan
13 5 10 Y Tyrosine
34 8 16 R Arginine
7 8 16 D Aspartic acid
24 4 18 G Glycine
9 1 10 H Histidine
5 5 5 M Methionine
18 8 16 F Phenylalanine
11 5 20 P Proline
15 7 9 Q Glutamine
20 10 17 E Glutamic acid 

8ITM_1|Chain A[auth R]|Gastric inhibitory polypeptide receptor|Homo sapiens (9606)
>5EWV_1|Chain A|Peregrin|Homo sapiens (9606)
>9ESJ_1|Chains A, C|Cyclin-dependent kinase 2|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 \(w=f(c)\)
8ITM , Knot 156 360 0.85 40 199 342 
MNSAHCNFRLPGSSDSPASASREAGITEAGITVAAGFVLRQCGSDGQWGLWRDHTQCENPEKNEAFLDQRLILERLQVMYTVGYSLSLATLLLALLILSLFRRLHCTRNYIHINLFTSFMLRAAAILSRDRLLPRPGPYLGDQALALWNQALAACRTAQIVTQYCVGANYTWLLVEGVYLHSLLVLVGGSEEGHFRYYLLLGWGAPALFVIPWVIVRYLYENTQCWERNEVKAIWWIIRTPILMTILINFLIFIRILGILLSKLRTRQMRCRDYRLRLARSTLFLVPLLGVHEVVFAPVTEEQARGALRFAKLGFEIFLSSFQGFLVSVLYCFINKEVQSEIRRGWHHCRLRRSLGEEQR
5EWV , Knot 62 116 0.84 38 95 112 
SMEMQLTPFLILLRKTLEQLQEKDTGNIFSEPVPLSEVPDYLDHIKKPMDFFTMKQNLEAYRYLNFDDFEEDFNLIVSNCLKYNAKDTIFYRAAVRLREQGGAVLRQARRQAEKMG
9ESJ , Knot 136 302 0.85 40 193 288 
GPGSMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL

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 \(c\).

\(|P_{f(8ITM_1)}(2) \setminus P_{f(5EWV_1)}(2)|=151\), \(|P_{f(5EWV_1)}(2) \setminus P_{f(8ITM_1)}(2)|=47\). 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 1:100100010111000011010001110011101111111000100101111000000001000011100011100101100110010110111111110110010000001010110011101111100001110111011001111100111100010110000111000111101101001111111000101000111111111111111111001000000100001011111100111101110111110111111001000010000001011000111111111001111110000101110110111011100101111011001100010001001100001000110000
Pair \(Z_2\) Length of longest common subsequence
8ITM_1,5EWV_1 198 4
8ITM_1,9ESJ_1 174 4
5EWV_1,9ESJ_1 168 3

Newick tree

 
[
	8ITM_1:96.06,
	[
		9ESJ_1:84,5EWV_1:84
	]:12.06
]

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 \((0.85)(0.8)(\frac{476 }{\log_{20} 476}-\frac{116}{\log_{20}116})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 8ITM_1 5EWV_1 133 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]