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Parikh vectors
6FDI_1 7PDW_1 5AUX_1 Letter Amino acid
26 11 18 V Valine
21 13 16 A Alanine
15 10 11 P Proline
41 12 30 L Leucine
23 10 15 T Threonine
3 5 1 C Cysteine
21 8 22 I Isoleucine
20 13 11 Q Glutamine
12 10 15 G Glycine
17 10 23 K Lycine
21 33 16 S Serine
15 10 13 R Arginine
17 12 12 N Asparagine
17 0 12 H Histidine
12 4 4 M Methionine
14 11 16 F Phenylalanine
5 2 2 W Tryptophan
8 8 10 Y Tyrosine
29 17 17 D Aspartic acid
27 7 29 E Glutamic acid 

6FDI_1|Chains A, B, C, D|cAMP-specific 3',5'-cyclic phosphodiesterase 4D|Homo sapiens (9606)
>7PDW_1|Chains A[auth AAA], E[auth FFF]|T-cell receptor alpha chain (TRAV/TRAC)|Homo sapiens (9606)
>5AUX_1|Chain A|Death-associated protein kinase 1|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 \(w=f(c)\)
6FDI , Knot 160 364 0.86 40 218 343 
GSHMIPRFGVKTEQEDVLAKELEDVNKWGLHVFRIAELSGNRPLTVIMHTIFQERDLLKTFKIPVDTLITYLMTLEDHYHADVAYHNNIHAADVVQSTHVLLSTPALEAVFTDLEILAAIFASAIHDVDHPGVSNQFLINTNSELALMYNDSSVLENHHLAVGFKLLQEENCDIFQNLTKKQRQSLRKMVIDIVLATDMSKHMNLLADLKTMVETKKVTSSGVLLLDNYSDRIQVLQNMVHCADLSNPTKPLQLYRQWTDRIMEEFFRQGDRERERGMEISPMCDKHNASVEKSQVGFIDYIVHPLWETWADLVHPDAQDILDTLEDNREWYQSTIPQSPSPAPDDPEEGRQGQTEKFQFELTL
7PDW , Knot 97 206 0.83 38 140 200 
MQKEVEQNSGPLSVPEGAIASLNCTYSDRGSQSFFWYRQYSGKSPELIMSIYSNGDKEDGRFTAQLNKASQYVSLLIRDSQPSDSATYLCAVRGTGRRALTFGSGTRLQVQPNIQNPDPAVYQLRDSKSSDKSVCLFTDFDSQTNVSQSKDSDVYITDKCVLDMRSMDFKSNSAVAWSNKSDFACANAFNNSIIPEDTFFPSPESS
5AUX , Knot 123 293 0.79 40 178 280 
MTVFRQENVDDYYDTGEELGSGQFAVVKKCREKSTGLQYAAKFIKKRRTKSSRRGVSREDIEREVSILKEIQHPNVITLHEVYENKTDVILILELVAGGELFDFLAEKESLTEEEATEFLKQILNGVYYLHSLQIAHFDLKPENIMLLDRNVPKPRIKIIDFGLAHKIDFGNEFKNIFGTPEFVAPEIVNYEPLGLEADMWSIGVITYILLSGASPFLGDTKQETLANVSAVNYEFEDEYFSNTSALAKDFIRRLLVKDPKKRMTIQDSLQHPWIKPKDTQQALSLEHHHHHH

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 \(c\).

\(|P_{f(6FDI_1)}(2) \setminus P_{f(7PDW_1)}(2)|=131\), \(|P_{f(7PDW_1)}(2) \setminus P_{f(6FDI_1)}(2)|=53\). 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 1:1001110111000000111001001001110110110101001101110011000011001011100110011010000010110000101101100001110011101110010111111101100100111000111000001111000001100001111101100000011001000000010011101111001000101110100110000100011111000000101100110010100100110100010001100110010000001101011000001010000111100110111001101101010011001000001000011001011100100100100001010101
Pair \(Z_2\) Length of longest common subsequence
6FDI_1,7PDW_1 184 4
6FDI_1,5AUX_1 172 3
7PDW_1,5AUX_1 176 3

Newick tree

 
[
	7PDW_1:91.32,
	[
		6FDI_1:86,5AUX_1:86
	]:5.32
]

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 \((0.85)(0.8)(\frac{570 }{\log_{20} 570}-\frac{206}{\log_{20}206})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 6FDI_1 7PDW_1 136 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]