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Parikh vectors
8TVM_1 2PHR_1 3KEW_1 Letter Amino acid
21 18 10 A Alanine
10 9 13 R Arginine
21 15 14 D Aspartic acid
27 9 26 E Glutamic acid
9 7 8 Y Tyrosine
21 30 6 S Serine
1 5 2 W Tryptophan
15 16 8 N Asparagine
11 15 8 Q Glutamine
18 18 21 G Glycine
19 10 16 I Isoleucine
23 9 24 K Lycine
5 0 5 C Cysteine
28 19 24 L Leucine
12 16 10 F Phenylalanine
9 12 5 P Proline
17 20 9 T Threonine
9 3 13 H Histidine
10 1 3 M Methionine
21 20 16 V Valine 

8TVM_1|Chains A, B, C, D|Interleukin-1 receptor-associated kinase 4|Homo sapiens (9606)
>2PHR_1|Chains A, B|Lectin|Pterocarpus angolensis (182271)
>3KEW_1|Chains A, B|DHHA1 domain protein|Clostridium perfringens (195103)
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 \(w=f(c)\)
8TVM , Knot 135 307 0.84 40 193 294 
ENKSLEVSDTRFHSFSFYELKNVTNNFDERPISVGGNKMGEGGFGVVYKGYVNNTTVAVKKLAAMVDITTEELKQQFDQEIKVMAKCQHENLVELLGFSSDGDDLCLVYVYMPNGSLLDRLSCLDGTPPLSWHMRCKIAQGAANGINFLHENHHIHRDIKSANILLDEAFTAKISDFGLARASEKFAQTVMTSRIVGTTAYMAPEALRGEITPKSDIYSFGVVLLEIITGLPAVDEHREPQLLLDIKEEIEDEEKTIEDYIDKKMNDADSTSVEAMYSVASQCLHEKKNKRPDIKKVQQLLQEMTAS
2PHR , Knot 115 252 0.84 38 172 243 
QDSLSFGFPTFPSDQKNLIFQGDAQIKNNAVQLTKTDSNGNPVASTVGRILFSAQVHLWEKSSSRVANFQSQFSFSLKSPLSNGADGIAFFIAPPDTTIPSGSGGGLLGLFAPGTAQNTSANQVIAVEFDTFYAQDSNTWDPNYPHIGIDVNSIRSVKTVKWDRRDGQSLNVLVTFNPSTRNLDVVATYSDGTRYEVSYEVDVRSVLPEWVRVGFSAASGEQYQTHTLESWSFTSTLLYTAQKKGENLALEM
3KEW , Knot 107 241 0.81 40 151 227 
MSLTKLYYEDQYIKEFKGEIIEVKEIDGKFHVLLDQTAFFPGGGGQMGDLGLIDGIKVLDVYEEEGKVYHVLEKEPKKLKNLQCELDWERRFDGMQQHLGQHLLSGCFYDLFGANTCGFHLGKEISTVDIVGFLDEKTIREAEKEANRLIFENLEVKSYAPSKKELKKVKTRRALPKTDEEIRIVEIVGLDLNACCGVHPRNTRDLQVIKIRRWEKHKNATRIEYVAGNRAVSEGHHHHHH

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 \(c\).

\(|P_{f(8TVM_1)}(2) \setminus P_{f(2PHR_1)}(2)|=93\), \(|P_{f(2PHR_1)}(2) \setminus P_{f(8TVM_1)}(2)|=72\). 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 1:0000101000010010100100100010001101110011011111100101000011100111110100001000100010111000000110111100010010110101101011001001010111010100011011101101100000100010010111001101010011110100011001100011100101110110101010001001111110110111110000010111010001000000100010001001000010110011000100000001010010011001010
Pair \(Z_2\) Length of longest common subsequence
8TVM_1,2PHR_1 165 3
8TVM_1,3KEW_1 160 4
2PHR_1,3KEW_1 157 3

Newick tree

 
[
	8TVM_1:82.15,
	[
		3KEW_1:78.5,2PHR_1:78.5
	]:3.65
]

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 \((0.85)(0.8)(\frac{559 }{\log_{20} 559}-\frac{252}{\log_{20}252})=87.1\)
Status Protein1 Protein2 d d1/2
Query variables 8TVM_1 2PHR_1 109 100.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]