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Parikh vectors
8AXH_1 2POV_1 8PPL_1 Letter Amino acid
4 1 5 C Cysteine
11 11 15 Q Glutamine
11 13 17 E Glutamic acid
6 17 5 P Proline
29 18 13 S Serine
10 8 7 Y Tyrosine
11 10 9 N Asparagine
36 22 9 G Glycine
16 26 25 L Leucine
8 23 12 K Lycine
24 12 11 T Threonine
7 7 4 W Tryptophan
16 13 16 A Alanine
3 2 8 M Methionine
18 17 10 V Valine
8 7 7 R Arginine
7 19 14 D Aspartic acid
3 12 5 H Histidine
8 9 15 I Isoleucine
8 12 11 F Phenylalanine 

8AXH_1|Chain A[auth H]|Single-chain variable antibody (scFv) SM3|Mus musculus (10090)
>2POV_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>8PPL_1|Chain A[auth I3]|Eukaryotic translation initiation factor 3 subunit K|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 \(w=f(c)\)
8AXH , Knot 106 244 0.79 40 148 218 
QVQLQESGGGLVQPGGSMKLSCVASGFTFSNYWMNWVRQSPEKGLEWVAEIRLKSNNYATHYAESVKGRFTISRDDSKSSVYLQMNNLRAEDTGIYYCTGVGQFAYWGQGTTVTVSSSSGGGGSGGGGGSSGSSDIVVTQESALTTSPGETVTLTCRSSTGAVTTSNYANWVQEKPDHLFTGLIGGTNNRAPGVPARFSGSLIGDKAALTITGAQTEDEAIYFCALWYSNHWVFGGGTKLTVLG
2POV , Knot 112 259 0.80 40 175 248 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASF
8PPL , Knot 98 218 0.80 40 152 204 
MAMFEQMRANVGKLLKGIDRYNPENLATLERYVETQAKENAYDLEANLAVLKLYQFNPAFFQTTVTAQILLKALTNLPHTDFTLCKCMIDQAHQEERPIRQILYLGDLLETCHFQAFWQALDENMDLLEGITGFEDSVRKFICHVVGITYQHIDRWLLAEMLGDLSDSQLKVWMSKYGWSADESGQIFICSQEESIKPKNIVEKIDFDSVSSIMASSQ

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 \(c\).

\(|P_{f(8AXH_1)}(2) \setminus P_{f(2POV_1)}(2)|=73\), \(|P_{f(2POV_1)}(2) \setminus P_{f(8AXH_1)}(2)|=100\). 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 1:0101000111110111010100110110100011011000100110111010100000100010010101010000000010101001010001100001110110110100101000011110111110010001110000110001100101000000111000001011000100110111110000111111010101110011101011000001101011100001111110010111
Pair \(Z_2\) Length of longest common subsequence
8AXH_1,2POV_1 173 4
8AXH_1,8PPL_1 174 4
2POV_1,8PPL_1 173 4

Newick tree

 
[
	8PPL_1:86.83,
	[
		8AXH_1:86.5,2POV_1:86.5
	]:0.33
]

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 \((0.85)(0.8)(\frac{503 }{\log_{20} 503}-\frac{244}{\log_{20}244})=74.3\)
Status Protein1 Protein2 d d1/2
Query variables 8AXH_1 2POV_1 95 91.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]