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Parikh vectors
4PTG_1 2FOQ_1 2DTU_1 Letter Amino acid
22 19 0 D Aspartic acid
23 24 0 K Lycine
6 2 0 M Methionine
36 18 0 S Serine
29 17 0 V Valine
34 13 3 A Alanine
27 7 0 R Arginine
24 22 5 G Glycine
20 9 0 I Isoleucine
38 26 0 L Leucine
2 7 0 W Tryptophan
18 10 1 N Asparagine
9 1 6 C Cysteine
20 13 0 E Glutamic acid
19 11 0 Q Glutamine
15 12 0 H Histidine
19 12 0 F Phenylalanine
31 17 0 P Proline
30 12 3 T Threonine
19 8 0 Y Tyrosine 

4PTG_1|Chains A, B|Glycogen synthase kinase-3 beta|Homo sapiens (9606)
>2FOQ_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>2DTU_1|Chains A[auth E], C[auth G], E[auth I], G[auth K]|5'-D(*CP*GP*(3DR)P*CP*TP*TP*AP*TP*GP*AP*CP*AP*GP*CP*CP*GP*CP*G)-3'|
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 \(w=f(c)\)
4PTG , Knot 180 441 0.82 40 237 409 
MHSSHHHHHHSSGENLYFQGHMSGRPRTTSFAESCKPVQQPSAFGSMKVSRDKDGSKVTTVVATPGQGPDRPQEVSYTDTKVIGNGSFGVVYQAKLCDSGELVAIKKVLQDKRFKNRELQIMRKLDHCNIVRLRYFFYSSGEKKDEVYLNLVLDYVPETVYRVARHYSRAKQTLPVIYVKLYMYQLFRSLAYIHSFGICHRDIKPQNLLLDPDTAVLKLCDFGSAKQLVRGEPNVSYICSRYYRAPELIFGATDYTSSIDVWSAGCVLAELLLGQPIFPGDSGVDQLVEIIKVLGTPTREQIREMNPNYTEFKFPQIKAHPWTKVFRPRTPPEAIALCSRLLEYTPTARLTPLEACAHSFFDELRDPNVKLPNGRDTPALFNFTTQELSSNPPLATILIPPHARIQAAASTPTNATAASDANTGDRGQTNNAASASASNST
2FOQ , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
2DTU , Knot 10 18 0.53 10 14 16 
CGNCTTATGACAGCCGCG

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 \(c\).

\(|P_{f(4PTG_1)}(2) \setminus P_{f(2FOQ_1)}(2)|=117\), \(|P_{f(2FOQ_1)}(2) \setminus P_{f(4PTG_1)}(2)|=56\). 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 1:100000000000100101010101010000110000110010111010100000100100111011011001001000000111010111100101000101111001100001000010110010000110100110001000001010111001100100110000010001111010101001100110100111000010100111010011101001101001101010100100000011011111000000101101101110111101111100110011011011101000010010100001011010101100110100110111100011000101010110101001100100101011010001111010000100011110111110101011100100101100100100100001101010000
Pair \(Z_2\) Length of longest common subsequence
4PTG_1,2FOQ_1 173 4
4PTG_1,2DTU_1 231 3
2FOQ_1,2DTU_1 184 2

Newick tree

 
[
	2DTU_1:10.73,
	[
		4PTG_1:86.5,2FOQ_1:86.5
	]:23.23
]

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 \((0.85)(0.8)(\frac{701 }{\log_{20} 701}-\frac{260}{\log_{20}260})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 4PTG_1 2FOQ_1 157 122.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]