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Parikh vectors
5MRM_1 6ZXY_1 9BUP_1 Letter Amino acid
4 9 7 N Asparagine
16 16 3 D Aspartic acid
2 6 0 C Cysteine
21 32 8 E Glutamic acid
12 28 11 G Glycine
23 40 12 L Leucine
10 15 8 T Threonine
10 19 7 A Alanine
8 6 2 Q Glutamine
14 19 5 I Isoleucine
6 8 3 M Methionine
12 10 2 P Proline
7 28 12 R Arginine
1 2 0 W Tryptophan
27 24 10 V Valine
2 7 1 H Histidine
8 16 7 F Phenylalanine
20 18 4 S Serine
7 16 4 Y Tyrosine
18 21 27 K Lycine 

5MRM_1|Chain A|2-C-methyl-D-erythritol 4-phosphate cytidylyltransferase, chloroplastic|Arabidopsis thaliana (3702)
>6ZXY_1|Chain A[auth B]|tRNA (Guanine(10)-N2)-dimethyltransferase|Archaeoglobus fulgidus (2234)
>9BUP_1|Chain A[auth S1]|40S ribosomal protein S24|Plasmodium falciparum 3D7 (36329)
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 \(w=f(c)\)
5MRM , Knot 103 228 0.81 40 149 221 
MEKSVSVILLAGGQGKRMKMSMPKQYIPLLGQPIALYSFFTFSRMPEVKEIVVVCDPFFRDIFEEYEESIDVDLSFAIPGKERQDSVYSGLQEIDVNSELVCIHDSARPLVNTEDVEKVLKDGSAVGAAVLGVPAKATIKEVNSDSLVVKTLDRKTLWEMQTPQVIKPELLKKGFELVKSEGLEVTDDVSIVEYLKHPVYVSQGSYTNIKVTTPDDLLLAERILSEDS
6ZXY , Knot 144 340 0.82 40 189 320 
MASMTGGQQMGRGSMKFLFYLSADNLEIARKEVLVLAERYGWVEDYQFEERLLLLDYAGEKFFERLAYTNEVTKIYDICSVSELEQVFSEIPVYDRLCCVRVKGGKGKTALERKLGALLWKRGAKVSVSNPEIVYKVYIQDDKCYVGLLEFERDTRQFFLRRPDRRPFLMPSAIKPKLARALVNLTGVLEGETLLDPMCGTGSFLIEAGLMGINPIGIDFIEKIVRGCRVNLEYYGIEGSVLLGDAKNLPLRDESVRGIATDYPYLRSTKAAGTLDELYSKTSEEFERVLKKGGRAAIVTNIDVESFFSNFEIEMKTEERVHGSLTRRIYLLRRHHHHHH
9BUP , Knot 63 133 0.77 36 94 125 
MTDQFTIRVKKYMSNPLLRRKQFALEILHPNKGSVAKKEVKERLAKMYKLNNVNTIVLFGFKTLFGGGRTKGFGLIYKNVDAVKKFEKKYRLVREGLIDKETKAGRRASKELKNRRKKVRGTEKTKVSGAKKK

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 \(c\).

\(|P_{f(5MRM_1)}(2) \setminus P_{f(6ZXY_1)}(2)|=52\), \(|P_{f(6ZXY_1)}(2) \setminus P_{f(5MRM_1)}(2)|=92\). 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 1:100010111111101001010110001111101111001101001101001111001110011000000101010111110000001001100101000110100010111000010011001011111111111010100100001110010000110100101101011001101100011010001011001001101001000010100100111100110000
Pair \(Z_2\) Length of longest common subsequence
5MRM_1,6ZXY_1 144 4
5MRM_1,9BUP_1 143 4
6ZXY_1,9BUP_1 157 4

Newick tree

 
[
	6ZXY_1:76.55,
	[
		5MRM_1:71.5,9BUP_1:71.5
	]:5.05
]

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 \((0.85)(0.8)(\frac{568 }{\log_{20} 568}-\frac{228}{\log_{20}228})=96.8\)
Status Protein1 Protein2 d d1/2
Query variables 5MRM_1 6ZXY_1 119 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]