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Parikh vectors
6DIK_1 3XIM_1 1LYC_1 Letter Amino acid
19 17 4 K Lycine
4 14 38 S Serine
5 18 19 T Threonine
1 5 2 W Tryptophan
9 12 3 Y Tyrosine
2 11 4 H Histidine
10 43 28 L Leucine
7 13 12 N Asparagine
6 33 20 D Aspartic acid
1 13 14 Q Glutamine
6 27 14 E Glutamic acid
3 14 16 I Isoleucine
1 4 7 M Methionine
6 46 35 A Alanine
5 26 15 R Arginine
6 18 29 P Proline
2 22 18 F Phenylalanine
4 20 20 V Valine
14 0 8 C Cysteine
10 37 37 G Glycine 

6DIK_1|Chains A, B|Basic phospholipase A2 homolog bothropstoxin-1|Bothrops jararacussu (8726)
>3XIM_1|Chains A, B, C, D|D-XYLOSE ISOMERASE|Actinoplanes missouriensis (1866)
>1LYC_1|Chains A, B|Peroxidase|Coprinopsis cinerea (5346)
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 \(w=f(c)\)
6DIK , Knot 58 121 0.76 40 89 117 
SLFELGKMILQETGKNPAKSYGAYGCNCGVLGRGKPKDATDRCCYVHKCCYKKLTGCNPKKDRYSYSWKDKTIVCGENNPCLKELCECDKAVAICLRENLGTYNKKYRYHLKPFCKKADPC
3XIM , Knot 162 393 0.82 38 205 370 
SVQATREDKFSFGLWTVGWQARDAFGDATRTALDPVEAVHKLAEIGAYGITFHDDDLVPFGSDAQTRDGIIAGFKKALDETGLIVPMVTTNLFTHPVFKDGGFTSNDRSVRRYAIRKVLRQMDLGAELGAKTLVLWGGREGAEYDSAKDVSAALDRYREALNLLAQYSEDRGYGLRFAIEPKPNEPRGDILLPTAGHAIAFVQELERPELFGINPETGHEQMSNLNFTQGIAQALWHKKLFHIDLNGQHGPKFDQDLVFGHGDLLNAFSLVDLLENGPDGAPAYDGPRHFDYKPSRTEDYDGVWESARANIRMYLLLRERARAFRADPEVQEALAASKVAELKTPTLNPGEGYAELLADRSAFEDYDADAVGAKGFGFVKLNQLAIEHLLGAR
1LYC , Knot 141 343 0.80 40 189 317 
QGPGGGGSVTCPGGQSTSNSQCCVWFDVLDDLQTNFYQGSKCESPVRKILRIVFHDAIGFSPALTAAGQFGGGGADGSIIAHSNIELAFPANGGLTDTVEALRAVGINHGVSFGDLIQFATAVGMSNCPGSPRLEFLTGRSSSSQPSPPSLIPGPGNTVTAILDRMGDAGFSPDEVVDLLAAHSLASQEGLNSAIFRSPLDSTPQVFDTQFYIETLLKGTTQPGPSLGFAEELSPFPGEFRMRSDALLARDSRTACRWQSMTSSNEVMGQRYRAAMAKMSVLGFDRNALTDCSDVIPSAVSNNAAPVIPGGLTVDDIEVSCPSEPFPEIAAASGPLPALAPAP

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 \(c\).

\(|P_{f(6DIK_1)}(2) \setminus P_{f(3XIM_1)}(2)|=39\), \(|P_{f(3XIM_1)}(2) \setminus P_{f(6DIK_1)}(2)|=155\). 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 1:0110110111000100110001101000111101010010000001000000010100100000000100001101000101001000001111010001100000000010110001010
Pair \(Z_2\) Length of longest common subsequence
6DIK_1,3XIM_1 194 4
6DIK_1,1LYC_1 206 3
3XIM_1,1LYC_1 150 4

Newick tree

 
[
	6DIK_1:10.09,
	[
		3XIM_1:75,1LYC_1:75
	]:32.09
]

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 \((0.85)(0.8)(\frac{514 }{\log_{20} 514}-\frac{121}{\log_{20}121})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 6DIK_1 3XIM_1 146 94.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]