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Parikh vectors
4RMD_1 5WBR_1 6WWN_1 Letter Amino acid
8 9 16 N Asparagine
20 15 37 E Glutamic acid
13 28 23 S Serine
14 32 34 V Valine
12 19 21 R Arginine
11 30 36 G Glycine
13 14 19 K Lycine
14 13 20 P Proline
4 3 19 Y Tyrosine
4 9 12 C Cysteine
12 20 27 D Aspartic acid
6 16 16 Q Glutamine
8 12 13 H Histidine
8 4 10 M Methionine
10 14 20 F Phenylalanine
10 9 29 T Threonine
21 23 36 A Alanine
33 27 32 L Leucine
2 4 4 W Tryptophan
12 12 27 I Isoleucine 

4RMD_1|Chain A|Retinoic acid receptor RXR-alpha|Homo sapiens (9606)
>5WBR_1|Chains A, B|Ketohexokinase|Homo sapiens (9606)
>6WWN_1|Chain A|Tubulin alpha-1B chain|Sus scrofa (9823)
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 \(w=f(c)\)
4RMD , Knot 109 235 0.84 40 167 223 
EDMPVERILEAELAVEPKTETYVEANMGLNPSSPNDPVTNICQAADKQLFTLVEWAKRIPHFSELPLDDQVILLRAGWNELLIASFSHRSIAVKDGILLATGLHVHRNSAHSAGVGAIFDRVLTELVSKMRDMQMDKTELGCLRAIVLFNPDSKGLSNPAEVEALREKVYASLEAYCKHKYPEQPGRFAKLLLRLPALRSIGLKCLEHLFFFKLIGDTPIDTFLMEMLEAPHQMT
5WBR , Knot 135 313 0.82 40 188 294 
MGSSHHHHHHSSGLVPRGSQILCVGLVVLDVISLVDKYPKEDSEIRCLSQRWQRGGNASNSCTVLSLLGAPCAFMGSMAPGHVADFLVADFRRRGVDVSQVAWQSKGDTPSSCCIINNSNGNRTIVLHDTSLPDVSATDFEKVDLTQFKWIHIEGRNASEQVKMLQRIDAHNTRQPPEQKIRVSVEVEKPREELFQLFGYGDVVFVSKDVAKHLGFQSAEEALRGLYGRVRKGAVLVCAWAEEGADALGPDGKLLHSDAFPPPRVVDTLGAGDTFNASVIFSLSQGRSVQEALRFGCQVAGKKCGLQGFDGIV
6WWN , Knot 192 451 0.86 40 258 426 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLISQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRSIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY

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 \(c\).

\(|P_{f(4RMD_1)}(2) \setminus P_{f(5WBR_1)}(2)|=73\), \(|P_{f(5WBR_1)}(2) \setminus P_{f(4RMD_1)}(2)|=94\). 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 1:0011100110101110100000101011101001001100100110001101101100110100111000111101110011110100001110011111011010000100111111100110011001001010000110101111101000110011010110001010101000000100110110111011110011100100111101110011001110110110010
Pair \(Z_2\) Length of longest common subsequence
4RMD_1,5WBR_1 167 4
4RMD_1,6WWN_1 173 4
5WBR_1,6WWN_1 158 5

Newick tree

 
[
	4RMD_1:86.92,
	[
		5WBR_1:79,6WWN_1:79
	]:7.92
]

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 \((0.85)(0.8)(\frac{548 }{\log_{20} 548}-\frac{235}{\log_{20}235})=89.3\)
Status Protein1 Protein2 d d1/2
Query variables 4RMD_1 5WBR_1 113 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]