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Parikh vectors
3JAI_1 4CIX_1 4FJP_1 Letter Amino acid
1 1 6 W Tryptophan
8 13 19 E Glutamic acid
32 13 25 G Glycine
8 8 20 S Serine
9 6 22 T Threonine
10 7 6 H Histidine
10 20 31 L Leucine
2 0 1 M Methionine
25 19 32 A Alanine
27 10 12 R Arginine
4 1 19 C Cysteine
4 4 13 Q Glutamine
16 10 5 I Isoleucine
7 1 10 F Phenylalanine
14 5 11 P Proline
22 13 26 V Valine
8 5 19 N Asparagine
7 5 21 D Aspartic acid
24 1 27 K Lycine
6 4 10 Y Tyrosine 

3JAI_1|Chain A|uL2|Oryctolagus cuniculus (9986)
>4CIX_1|Chain A|3-DEHYDROQUINATE DEHYDRATASE|MYCOBACTERIUM TUBERCULOSIS (1773)
>4FJP_1|Chain A|Lactotransferrin|Bos taurus (9913)
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 \(w=f(c)\)
3JAI , Knot 111 244 0.83 40 157 233 
GRVIRGQRKGAGSVFRAHVKHRKGAARLRAVDFAERHGYIKGIVKDIIHDPGRGAPLAKVVFRDPYRFKKRTELFIAAEGIHTGQFVYCGKKAQLNIGNVLPVGTMPEGTIVCCLEEKPGDRGKLARASGNYATVISHNPETKKTRVKLPSGSKKVISSANRAVVGVVAGGGRIDKPILKAGRAYHKYKAKRNCWPRVRGVAMNPVEHPFGGGNHQHIGKPSTIRRDAPAGRKVGLIAARRTGR
4CIX , Knot 66 146 0.75 38 102 138 
SELIVNVINGPNLGRLGRREPAVYGGTTHDELVALIEREAAELGLKAVVRQSDSEAQLLDWIHQAADAAEPVILNAGGLTHTSVALRDACAELSAPLIEVHISNVHAREEFRRHSYLSPIATGVIVGLGIQGYLLALRYLAEHVGT
4FJP , Knot 141 335 0.81 40 194 317 
YTRVVWCAVGPEEQKKCQQWSQQSGQNVTCATASTTDDCIVLVLKGEADALNLDGGYIYTAGKCGLVPVLAENRKSSKHSSLDCVLRPTEGYLAVAVVKKANEGLTWNSLKDKKSCHTAVDRTAGWNIPMGLIVNQTGSCAFDEFFSQSCAPGADPKSRLCALCAGDDQGLDKCVPNSKEKYYGYTGAFRCLAEDVGDVAFVKNDTVWENTNGESTADWAKNLKREDFRLLCLDGTRKPVTEAQSCHLAVAPNHAVVSRSDRAAHVEQVLLHQQALFGKNGKNCPDKFCLFKSETKNLLFNDNTECLAKLGGRPTYEEYLGTEYVTAIANLKKCS

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 \(c\).

\(|P_{f(3JAI_1)}(2) \setminus P_{f(4CIX_1)}(2)|=101\), \(|P_{f(4CIX_1)}(2) \setminus P_{f(3JAI_1)}(2)|=46\). 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 1:1011010001110110101000011101011011000101011100110011011111011100100100000111110110010110010010101101111101101011001000110010110101001011000100000010110100011001001111111111010011101101000001000011010111101100111110000110100100011110011111100010
Pair \(Z_2\) Length of longest common subsequence
3JAI_1,4CIX_1 147 3
3JAI_1,4FJP_1 179 3
4CIX_1,4FJP_1 174 4

Newick tree

 
[
	4FJP_1:92.65,
	[
		3JAI_1:73.5,4CIX_1:73.5
	]:19.15
]

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 \((0.85)(0.8)(\frac{390 }{\log_{20} 390}-\frac{146}{\log_{20}146})=73.4\)
Status Protein1 Protein2 d d1/2
Query variables 3JAI_1 4CIX_1 91 72.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]