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Parikh vectors
1ZKI_1 3TQU_1 6CTN_1 Letter Amino acid
3 8 0 K Lycine
7 3 0 M Methionine
0 1 0 W Tryptophan
2 3 0 Y Tyrosine
14 27 4 A Alanine
6 11 0 D Aspartic acid
4 5 0 H Histidine
13 19 0 L Leucine
3 2 7 C Cysteine
16 11 4 G Glycine
11 16 0 S Serine
3 9 0 N Asparagine
5 9 0 Q Glutamine
3 9 1 T Threonine
3 10 0 P Proline
15 10 0 V Valine
11 9 0 R Arginine
8 20 0 E Glutamic acid
3 13 0 I Isoleucine
3 8 0 F Phenylalanine 

1ZKI_1|Chains A, B|hypothetical protein PA5202|Pseudomonas aeruginosa (208964)
>3TQU_1|Chains A, B, C, D|Non-canonical purine NTP pyrophosphatase|Coxiella burnetii (227377)
>6CTN_1|Chain A[auth T]|DNA (5'-D(*CP*CP*GP*AP*CP*AP*GP*CP*GP*CP*AP*TP*CP*AP*GP*C)-3')|Homo sapiens (9606)
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 \(w=f(c)\)
1ZKI , Knot 67 133 0.82 38 107 128 
NGMSEMPAREQMISAYSELVGLDPVSLGDGVAEVRLPMAAHLRNRGGVMHGGALFSLMDVTMGLACSSSHGFDRQSVTLECKINYIRAVADGEVRCVARVLHAGRRSLVVEAEVRQGDKLVAKGQGTFAQLGS
3TQU , Knot 92 203 0.80 40 143 193 
SNAMLEIVLASQNSSKLAEMQELLRDLEIKFIPQTEFSVPDIEETGSTFVENAIIKARHAAKQTGLPALADDSGLTIAALNSAPGVFSSRYAGKNATDAERIQKVLEALEAADDSDRSASFHCVIALMENENDPAPLICHGVWEGEIAREPRGKNGFGYDPIFYVPSHQRTAAELDPQEKNAISHRGQALEQLSTVLTEAFLV
6CTN , Knot 8 16 0.46 8 9 12 
CCGACAGCGCATCAGC

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 \(c\).

\(|P_{f(1ZKI_1)}(2) \setminus P_{f(3TQU_1)}(2)|=62\), \(|P_{f(3TQU_1)}(2) \setminus P_{f(1ZKI_1)}(2)|=98\). 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 1:0110011100011010001111011011011101011111010001111011111011010111100000110000101000100101110101001101101100011101010010011101010110110
Pair \(Z_2\) Length of longest common subsequence
1ZKI_1,3TQU_1 160 3
1ZKI_1,6CTN_1 110 2
3TQU_1,6CTN_1 148 2

Newick tree

 
[
	3TQU_1:83.12,
	[
		1ZKI_1:55,6CTN_1:55
	]:28.12
]

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 \((0.85)(0.8)(\frac{336 }{\log_{20} 336}-\frac{133}{\log_{20}133})=62.2\)
Status Protein1 Protein2 d d1/2
Query variables 1ZKI_1 3TQU_1 80 65.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]