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Parikh vectors
8GTM_1 1JPV_1 6ZIH_1 Letter Amino acid
14 14 17 N Asparagine
22 9 17 F Phenylalanine
7 6 7 P Proline
14 8 10 R Arginine
17 2 1 H Histidine
30 25 20 L Leucine
9 16 26 K Lycine
13 11 19 Y Tyrosine
26 23 16 V Valine
11 16 20 E Glutamic acid
23 12 26 I Isoleucine
13 16 13 S Serine
7 3 5 W Tryptophan
14 17 27 T Threonine
28 19 10 A Alanine
4 8 20 D Aspartic acid
7 2 0 C Cysteine
5 3 14 Q Glutamine
13 22 23 G Glycine
7 4 7 M Methionine 

8GTM_1|Chain A|Isoform CRF-R2 of Corticotropin-releasing factor receptor 1|Homo sapiens (9606)
>1JPV_1|Chains A, B, C|5'-deoxy-5'-methylthioadenosine phosphorylase|Sulfolobus solfataricus (2287)
>6ZIH_1|Chains AA[auth 3], A[auth 2], BA[auth 5], B[auth 4], CA[auth 7], C[auth 6], DA[auth T], D[auth S], EA[auth V], E[auth U], F[auth W], Z[auth X]|Distal tail protein|Lactococcus phage p2 (254252)
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 \(w=f(c)\)
8GTM , Knot 124 284 0.82 40 180 266 
MEILNEEKKSKVHYHVAAIINYLGHCISLVALLVAFVLFLRARSIRCLRNIIHANLIAAFILRNATWFVVQLTMSPEVHQSNVGWCRLVTAAYNYFHVTNFFWMFGEGCYLHTAIVLTYSTDRLRAWMFICIGWGVPFPIIVAWAIGKLYYDNEKCWAGKRPGVYTDYIYQGPMALVLLINFIFLFNIVRILMTKLRASTTSETIQARKAVKATLVLLPLLGITYMLAFVNPGEDEVSRVVFIYFNAFLESFQGFFVSVFACFLNSEVRSAAAAHHHHHHHHHH
1JPV , Knot 108 236 0.83 40 160 226 
MNPVHILAKKGEVAERVLVVGDPGRARLLSTLLQNPKLTNENRGFLVYTGKYNGETVSIATHGIGGPSIAIVLEELAMLGANVFIRYGTTGALVPYINLGEYIIVTGASYNQGGLFYQYLRDNACVASTPDFELTNKLVTSFSKRNLKYYVGNVFSSDAFYAEDEEFVKKWSSRGNIAVEMECATLFTLSKVKGWKSATVLVVSDNLAKGGIWITKEELEKSVMDGAKAVLDTLTS
6ZIH , Knot 134 298 0.85 38 180 281 
MVRQYKIHTNLDGTDDKVWDVTNGKVRFYQPSNLGLQSTNNIWQSNGIGVMGTRSITQPQIEFKLETFGESLEENYQLMKDFVNDILSKKFVTLEYQTEIFQVYADLALADVTKTEGYGKNGTFSEKITFDIITKWYTYENLTFDKIQNGKVIAGMSKIYGGTAPGNYKYIKGTSYTYYGESDIDRLSRWDIKEEIFSFMGILYPKLPKTPAGVRFLDDIGNEYTAIVFKTEQVQDYILINTDVNDETYQGWKGTTALNLFPVMDFERYRTRIIEKGQMELINLSKAEFKIKRKADFV

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 \(c\).

\(|P_{f(8GTM_1)}(2) \setminus P_{f(1JPV_1)}(2)|=90\), \(|P_{f(1JPV_1)}(2) \setminus P_{f(8GTM_1)}(2)|=70\). 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 1:10110000000100011111001100101111111111110100100100110101111111001011110101010100001110011011000101001111110100100111100000010111110111111111111111101000000011100111000010011111111101111101101110010100000010100110101111111110011111011000100111101011100101111011101100010011110000000000
Pair \(Z_2\) Length of longest common subsequence
8GTM_1,1JPV_1 160 4
8GTM_1,6ZIH_1 182 4
1JPV_1,6ZIH_1 158 3

Newick tree

 
[
	8GTM_1:87.78,
	[
		1JPV_1:79,6ZIH_1:79
	]:8.78
]

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 \((0.85)(0.8)(\frac{520 }{\log_{20} 520}-\frac{236}{\log_{20}236})=81.3\)
Status Protein1 Protein2 d d1/2
Query variables 8GTM_1 1JPV_1 103 94.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]