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Parikh vectors
4QQJ_1 7TUB_1 8YMU_1 Letter Amino acid
7 31 17 T Threonine
21 22 29 R Arginine
20 26 22 D Aspartic acid
6 8 12 C Cysteine
41 32 31 L Leucine
10 17 10 M Methionine
24 30 28 A Alanine
24 35 20 G Glycine
8 16 19 Y Tyrosine
24 33 32 V Valine
5 4 3 W Tryptophan
22 36 27 E Glutamic acid
14 10 14 H Histidine
10 18 38 I Isoleucine
14 15 15 K Lycine
26 20 21 P Proline
7 21 18 N Asparagine
9 21 19 Q Glutamine
11 23 24 F Phenylalanine
20 26 32 S Serine 

4QQJ_1|Chain A|Fibroblast growth factor receptor 4|Homo sapiens (9606)
>7TUB_1|Chain A|Tubulin beta chain|Homo sapiens (9606)
>8YMU_1|Chains A, B, C|Broad-range thermal receptor 1|Scolopendra mutilans (2836329)
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 \(w=f(c)\)
4QQJ , Knot 139 323 0.82 40 195 302 
MGSSHHHHHHSQDPLLAGLVSLDLPLDPLWEFPRDRLVLGKPLGEGAFGQVVRAEAFGMDPARPDQASTVAVKMLKDNASDKDLADLVSEMEVMKLIGRHKNIINLLGVCTQEGPLYVILECAAKGNLREFLRARRPPGPDLSPDGPRSSEGPLSFPVLVSCAYQVARGMQYLESRKCIHRDLAARNVLVTEDNVMKIADFGLARGVHHIDYYKKTSNGRLPVKWMAPEALFDEVYTHQSDVWSFGILLWEIFTLGGSPYPGIPVEELFSLLREGHRMDRPPHCPPELYGLMRECWHAAPSQRPTFKQLVEALDKVLLAVSEE
7TUB , Knot 189 444 0.86 40 257 426 
MREIVHIQAGQCGNQIGAKFWEVISDEHGIDPTGTYHGDSDLQLDRISVYYNEATGGKYVPRAILVDLEPGTMDSVRSGPFGQIFRPDNFVFGQSGAGNNWAKGHYTEGAELVDSVLDVVRKEAESCDCLQGFQLTHSLGGGTGSGMGTLLISKIREEYPDRIMNTFSVVPSPKVSDTVVEPYNATLSVHQLVENTDETYCIDNEALYDICFRTLKLTTPTYGDLNHLVSATMSGVTTCLRFPGQLNADLRKLAVNMVPFPRLHFFMPGFAPLTSRGSQQYRALTVPELTQQVFDAKNMMAACDPRHGRYLTVAAVFRGRMSMKEVDEQMLNVQNKNSSYFVEWIPNNVKTAVCDIPPRGLKMAVTFIGNSTAIQELFKRISEQFTAMFRRKAFLHWYTGEGMDEMEFTEAESNMNDLVSEYQQYQDATAEEEEDFGEEAEEEA
8YMU , Knot 181 431 0.85 40 246 412 
MAPSAYALDRSSVPSGKPRLMEGAESKARAGARKIHHLKIHGFDRVFGPGTHRWQRALWFAGVVFCFGISAHQIAMHVLDYLSEPVAVRIDFVAQNELRIPEITVCPRIFQQNTIFTDMVEKQGIDKMKFLDLIDRPEYDIMAVWNLSRIFSDNVSCSAHEGSSIISGSYVDPNMSQPHLVYTTSGQCINIPASRPLIYRGVNTFVRITDSQPRNLDEVRPEAIQIYFHEHHHAHLSRYLTGLRGYIVPIANPFAFSIRFTQINYANRTDSPCVDSEEYAACVEDFIEQRIYEKAQVQCRLPYMRPKLPLCSTPTDARKIFVATDDVIQNFEKESSCKRKCEENLYIVEFMHLFERSNISIDMSVYFAYNYIQVATEYLTYTLRGLLSDIGGVLGLFLGICILSVIEVFEVVIFSLQRVLCNEDKKQQNNE

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 \(c\).

\(|P_{f(4QQJ_1)}(2) \setminus P_{f(7TUB_1)}(2)|=48\), \(|P_{f(7TUB_1)}(2) \setminus P_{f(4QQJ_1)}(2)|=110\). 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 1:11000000000001111111010111011101100011110111011110110101111011010010011101100010000110110010110111000011011110000111011100110101001101001111010101100001110111110010011011001000001000111001110000110110111101100100000000101110111101110010000001101111110110111010111110011011001001001100110101110001011100010100110110011111000
Pair \(Z_2\) Length of longest common subsequence
4QQJ_1,7TUB_1 158 5
4QQJ_1,8YMU_1 175 3
7TUB_1,8YMU_1 149 4

Newick tree

 
[
	4QQJ_1:86.10,
	[
		7TUB_1:74.5,8YMU_1:74.5
	]:11.60
]

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 \((0.85)(0.8)(\frac{767 }{\log_{20} 767}-\frac{323}{\log_{20}323})=121.\)
Status Protein1 Protein2 d d1/2
Query variables 4QQJ_1 7TUB_1 159 134.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]