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Parikh vectors
5XSI_1 4YGT_1 5DNW_1 Letter Amino acid
8 7 13 P Proline
10 10 7 Q Glutamine
19 17 23 G Glycine
27 5 17 I Isoleucine
27 14 12 T Threonine
1 1 2 W Tryptophan
27 12 30 V Valine
25 6 17 R Arginine
10 5 11 H Histidine
33 12 23 L Leucine
9 9 10 Y Tyrosine
26 10 11 E Glutamic acid
6 6 13 F Phenylalanine
21 5 24 S Serine
0 0 2 C Cysteine
13 8 4 K Lycine
10 3 9 M Methionine
32 10 20 A Alanine
14 11 11 N Asparagine
25 9 16 D Aspartic acid 

5XSI_1|Chains A, B|Phosphodiesterase acting on cyclic dinucleotides|Staphylococcus aureus (1280)
>4YGT_1|Chain A|Uncharacterized protein YjgB|Bacillus subtilis (224308)
>5DNW_1|Chain A|ShKAI2iB|Striga hermonthica (68872)
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 \(w=f(c)\)
5XSI , Knot 142 343 0.80 38 188 320 
GSMRTRVRARVISHALKDILAEGDKVIIMGHKRPDLDAIGAAIGVSRFAMMNNLEAYIVLNETDIDPTLRRVMNEIDKKPELRERFITSDDAWDMMTSKTTVVIVDTHKPELVLDENVLNKANRKVVIDHHRRGESFISNPLLIYMEPYASSTAELVTELLEYQPTEQRLTRLESTVMYAGIIVDTRNFTLRTGSRTFDAASYLRAHGADTILTQHFLKDDVDTYINRSELIRTVKVEDNGIAIAHGSDDKIYHPVTVAQAADELLSLEGIEASYVVARREDNLIGISARSLGSVNVQLTMEALGGGGHLTNAATQLKGVTVEEAIAQLQQAITEQLSRSEDA
4YGT , Knot 76 160 0.80 38 124 158 
GAAQTNTLSENTNQSAAELVKNLYNTAYKGEMPQQAQGLTINKSTKGDVHAAFGEPERPVGGDNRFDLYHWNMGQPGYGFSYHKDMTISEIRYFGTGVERQLNLGGVTPEVLQKQLGPVNRVLTVPFTDEIDYVYDTGRYELHFVIGTDQTADHVNLKAK
5DNW , Knot 124 275 0.84 40 176 262 
GPLGSMNRVEAARNVHIVGSGDTTVVLGHGFGTDQSVWKHLVPYLVDSYRVLLYDNMGAGSTNPEYFHFERYSTLQGYAHDLLVILHEFNIRSCIFVGHSLSAMTGAIASIIRPDLFQKIVMLSASPRFLNTADYLGGFEPADVEQLAGAIEANYKSWVSGFAPMVVGGDMDSVAVQEFSRTLFNMRPDIARSVFRTIFTSDLRDYLGRVTVPCHIIQSSRDMAVPVSVAGYIHNRVGGRSVVEVMNTEGHLPQLSAPEVAIPVLLRHIKNDIDS

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 \(c\).

\(|P_{f(5XSI_1)}(2) \setminus P_{f(4YGT_1)}(2)|=113\), \(|P_{f(4YGT_1)}(2) \setminus P_{f(5XSI_1)}(2)|=49\). 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 1:1010001010110011001110100111110001010111111110011110010101110000101010011001000101000110000110110000011110000101110001100100011100000100110011110101010001011001100010000100100011011111000010100100010110010101100110001100010001000011001010001111101000010011011011001101011010011100000111101001101010101011111101001100101101001110100110001000001
Pair \(Z_2\) Length of longest common subsequence
5XSI_1,4YGT_1 162 4
5XSI_1,5DNW_1 166 4
4YGT_1,5DNW_1 156 4

Newick tree

 
[
	5XSI_1:83.29,
	[
		4YGT_1:78,5DNW_1:78
	]:5.29
]

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 \((0.85)(0.8)(\frac{503 }{\log_{20} 503}-\frac{160}{\log_{20}160})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 5XSI_1 4YGT_1 127 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]