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Parikh vectors
4TKM_1 3CQR_1 5HKL_1 Letter Amino acid
10 13 1 N Asparagine
29 13 20 G Glycine
20 11 6 I Isoleucine
13 12 1 F Phenylalanine
6 10 13 H Histidine
11 8 2 K Lycine
17 14 10 S Serine
2 4 1 W Tryptophan
5 10 4 Y Tyrosine
34 7 25 A Alanine
15 8 18 R Arginine
17 16 12 D Aspartic acid
2 2 0 C Cysteine
4 6 3 Q Glutamine
10 9 12 E Glutamic acid
7 11 4 P Proline
19 11 19 L Leucine
6 1 4 M Methionine
15 9 12 T Threonine
16 10 23 V Valine 

4TKM_1|Chains A, B|NADH-dependent reductase for 4-deoxy-L-erythro-5-hexoseulose uronate|Sphingomonas sp. A1 (90322)
>3CQR_1|Chains A, B|Violaxanthin de-epoxidase, chloroplast|Arabidopsis thaliana (3702)
>5HKL_1|Chains A, B|Orotate phosphoribosyltransferase|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
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 \(w=f(c)\)
4TKM , Knot 114 258 0.81 40 167 245 
MFSDLKGKRILITGSTEGIGMATAIELARYGAVVGLNSHVDPADPALLLGKLREAGGDGAFFRADITKTAECQRLVSAFVERFDGIDVLINNAGGLAGRSNLENIDDAFYDRVMDLNGRSVLMMTKFAIPHLRASAKASGTTSAVISTGSIAAREGGGIGAGVYAASKAWLHDIHRNWVKEFTKDSIRFNIVAPGTVDTAFHADKSDELKTRIANSIPMGRFGTVQELAPAYVFFASHAASGYITGQILDVNGGQICP
3CQR , Knot 91 185 0.85 40 144 176 
MDLGEFPAPDPSVLVQNFNISDFNGKWYITSGLNPTFDAFDCQLHEFHTEGDNKLVGNISWRIKTLDSGFFTRSAVQKFVQDPNQPGVLYNHDNEYLHYQDDWYILSSKIENKPEDYIFVYYRGRNDAWDGYGGAVVYTRSSVLPNSIIPELEKAAKSIGRDFSTFIRTDNTCGPEPRSHHHHHH
5HKL , Knot 80 190 0.73 38 108 174 
HHHHHHIEGRHMAGPDRAELAELVRRLSVVHGRVTLSSGREADYYVDLRRATLHHRASALIGRLMRELTADWDYSVVGGLTLGADPVATAIMHAPGRPIDAFVVRKSAKAHGMQRLIEGSEVTGQRVLVVEDTSTTGNSALTAVHAVQDVGGEVVGVATVVDRATGAAEAIEAEGLRYRSVLGLADLGLD

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 \(c\).

\(|P_{f(4TKM_1)}(2) \setminus P_{f(3CQR_1)}(2)|=89\), \(|P_{f(3CQR_1)}(2) \setminus P_{f(4TKM_1)}(2)|=66\). 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 1:110010100111010001111101101100111111000101101111110100111011110101000100001101110010110111001111110001001001100011010100111100111101010101010001110010111001111111101100111001000110010000101011111010011010000010001100111101101001111011110011010101011010110101
Pair \(Z_2\) Length of longest common subsequence
4TKM_1,3CQR_1 155 4
4TKM_1,5HKL_1 131 4
3CQR_1,5HKL_1 150 6

Newick tree

 
[
	3CQR_1:79.52,
	[
		4TKM_1:65.5,5HKL_1:65.5
	]:14.02
]

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 \((0.85)(0.8)(\frac{443 }{\log_{20} 443}-\frac{185}{\log_{20}185})=75.9\)
Status Protein1 Protein2 d d1/2
Query variables 4TKM_1 3CQR_1 93 79
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]