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Parikh vectors
4LYS_1 4MQR_1 1ZYB_1 Letter Amino acid
12 3 12 N Asparagine
2 8 6 C Cysteine
2 44 8 G Glycine
5 15 6 M Methionine
5 12 12 F Phenylalanine
12 26 7 P Proline
8 28 14 T Threonine
3 26 8 R Arginine
5 40 9 V Valine
1 24 13 H Histidine
13 46 30 L Leucine
12 4 18 K Lycine
3 9 1 W Tryptophan
6 25 14 D Aspartic acid
7 10 7 Y Tyrosine
7 19 20 I Isoleucine
9 12 9 Q Glutamine
8 15 15 E Glutamic acid
1 24 14 S Serine
5 67 9 A Alanine 

4LYS_1|Chain A|Bromodomain-containing protein 4|Homo sapiens (9606)
>4MQR_1|Chains A, B|Adenosylmethionine-8-amino-7-oxononanoate aminotransferase|Mycobacterium tuberculosis (1773)
>1ZYB_1|Chain A|transcription regulator, CRP family|Bacteroides thetaiotaomicron (226186)
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 \(w=f(c)\)
4LYS , Knot 65 126 0.83 40 109 122 
MNPPPPETSNPNKPKRQTNQLQYLLRVVLKTLWKHQFAWPFQQPVDAVKLNLPDYYKIIKTPMDMGTIKKRLENNYYWNAQECIQDFNTMFTNCYIYNKPGDDIVLMAEALEKLFLQKINELPTEE
4MQR , Knot 185 457 0.82 40 219 411 
MGSSHHHHHHSSGLVPRGSHMAAATGGLTPEQIIAVDGAHLWHPYSSIGREAVSPVVAVAAHGAWLTLIRDGQPIEVLDAMSSWWTAIHGHGHPALDQALTTQLRVMNHVMFGGLTHEPAARLAKLLVDITPAGLDTVFFSDSGSVSVEVAAKMALQYWRGRGLPGKRRLMTWRGGYHGDTFLAMSICDPHGGMHSLWTDVLAAQVFAPQVPRDYDPAYSAAFEAQLAQHAGELAAVVVEPVVQGAGGMRFHDPRYLHDLRDICRRYEVLLIFDEIATGFGRTGALFAADHAGVSPDIMCVGKALTGGYLSLAATLCTADVAHTISAGAAGALMHGPTFMANPLACAVSVASVELLLGQDWRTRITELAAGLTAGLDTARALPAVTDVRVCGAIGVIECDRPVDLAVATPAALDRGVWLRPFRNLVYAMPPYICTPAEITQITSAMVEVARLVGSLP
1ZYB , Knot 106 232 0.83 40 161 223 
MGSDKIHHHHHHMETMFDTLLQLPLFQGLCHEDFTSILDKVKLHFIKHKAGETIIKSGNPCTQLCFLLKGEISIVTNAKENIYTVIEQIEAPYLIEPQSLFGMNTNYASSYVAHTEVHTVCISKAFVLSDLFRYDIFRLNYMNIVSNRAQNLYSRLWDEPTLDLKSKIIRFFLSHCEKPQGEKTFKVKMDDLARCLDDTRLNISKTLNELQDNGLIELHRKEILIPDAQKLL

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 \(c\).

\(|P_{f(4LYS_1)}(2) \setminus P_{f(4MQR_1)}(2)|=55\), \(|P_{f(4MQR_1)}(2) \setminus P_{f(4LYS_1)}(2)|=165\). 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 1:101111000010010000001001101110011000111110011011010110000110011011010001000001010001001001100001000110011111011001110010011000
Pair \(Z_2\) Length of longest common subsequence
4LYS_1,4MQR_1 220 3
4LYS_1,1ZYB_1 158 3
4MQR_1,1ZYB_1 190 6

Newick tree

 
[
	4MQR_1:10.55,
	[
		4LYS_1:79,1ZYB_1:79
	]:30.55
]

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 \((0.85)(0.8)(\frac{583 }{\log_{20} 583}-\frac{126}{\log_{20}126})=133.\)
Status Protein1 Protein2 d d1/2
Query variables 4LYS_1 4MQR_1 166 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]