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Parikh vectors
4XKY_1 4WPU_1 5QXS_1 Letter Amino acid
12 43 16 D Aspartic acid
18 38 6 P Proline
19 29 6 S Serine
1 20 0 W Tryptophan
9 31 14 R Arginine
8 19 3 H Histidine
36 50 13 L Leucine
11 26 5 F Phenylalanine
12 36 6 T Threonine
12 16 5 Y Tyrosine
5 9 3 C Cysteine
23 60 1 G Glycine
18 24 11 I Isoleucine
13 28 7 K Lycine
24 40 7 V Valine
11 14 3 Q Glutamine
13 22 3 N Asparagine
18 42 13 E Glutamic acid
11 19 2 M Methionine
26 56 6 A Alanine 

4XKY_1|Chain A|Dihydrodipicolinate synthase|Bacteroides thetaiotaomicron (226186)
>4WPU_1|Chain A|Phosphoenolpyruvate carboxykinase [GTP]|Mycobacterium tuberculosis (419947)
>5QXS_1|Chain A|ATPase family AAA domain-containing protein 2|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 \(w=f(c)\)
4XKY , Knot 134 300 0.85 40 190 288 
MNIPLSGIIPPLVTPLLDDDVLDVEGLQRLIEHLIAGGVHALFVLGTTGESQSLSYKLRMEMIKNTCRIAKGRLPVLVCISDTSIVESVNLACLAADHGADAVVSAPPYYFATGQPELIEFYEHLLPQLPLPLFLYNMPTHTKVNFAPATIQRIAENPGVIGFKDSSANTVYFQSVMYAMKDNPDFSMLVGPEEIMAESVLLGAHGGVNGGANMFPELYVSLYNAAKNADMEEVRRLQEKVMQISATIYTVGQHGSSYLKGLKCALSLLGICSDYVAAPFHKFEQRERGKIWKALQNLGV
4WPU , Knot 251 622 0.86 40 280 583 
HHHHHHSSGLVPAGSHMTSATIPGLDTAPTNHQGLLSWVEEVAELTQPDRVVFTDGSEEEFQRLCDQLVEAGTFIRLNPEKHKNSYLALSDPSDVARVASRTYICSAKEIDAGPTNNWMDPGEMRSIMKDLYRGCMRGRTMYVVPFCMGPLGAEDPKLGVEITDSEYVVVSMRTMTRMGKAALEKMGDDGFFVKALHSVGAPLEPGQKDVAWPCSETKYITHFPETREIWSYGSGYGGNALLGKKCYSLRIASAMAHDEGWLAEHMLILKLISPENKAYYFAAAFPSACGKTNLAMLQPTIPGWRAETLGDDIAWMRFGKDGRLYAVNPEFGFFGVAPGTNWKSNPNAMRTIAAGNTVFTNVALTDDGDVWWEGLEGDPQHLIDWKGNDWYFRETETNAAHPNSRYCTPMSQCPILAPEWDDPQGVPISGILFGGRRKTTVPLVTEARDWQHGVFIGATLGSEQTAAAEGKVGNVRRDPMAMLPFLGYNVGDYFQHWINLGKHADESKLPKVFFVNWFRRGDDGRFLWPGFGENSRVLKWIVDRIEHKAGGATTPIGTVPAVEDLDLDGLDVDAADVAAALAVDADEWRQELPLIEEWLQFVGEKLPTGVKDEFDALKERLG
5QXS , Knot 63 130 0.78 38 97 125 
SMQEEDTFRELRIFLRNVTHRLAIDKRFRVFTKPVDPDEVPDYRTVIKEPMDLSSVISKIDLHKYLTVKDYLRDIDLICSNALEYNPDRDPGDRLIRHRACALRDTAYAIIKEELDEDFEQLCEEIQESR

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 \(c\).

\(|P_{f(4XKY_1)}(2) \setminus P_{f(4WPU_1)}(2)|=46\), \(|P_{f(4WPU_1)}(2) \setminus P_{f(4XKY_1)}(2)|=136\). 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 1:101110111111101110001101011001100111111011111100100001000101011000001101011111010000110010110111001101110111001101010110100011101111111001100001011110100110011111100001001010011011000101011111001110011111011101110111010101001100101001001000110101010011001000101100110111100001111100100000101101100111
Pair \(Z_2\) Length of longest common subsequence
4XKY_1,4WPU_1 182 4
4XKY_1,5QXS_1 187 3
4WPU_1,5QXS_1 221 3

Newick tree

 
[
	5QXS_1:10.86,
	[
		4XKY_1:91,4WPU_1:91
	]:14.86
]

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 \((0.85)(0.8)(\frac{922 }{\log_{20} 922}-\frac{300}{\log_{20}300})=167.\)
Status Protein1 Protein2 d d1/2
Query variables 4XKY_1 4WPU_1 215 157.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]