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Parikh vectors
7MUC_1 8GST_1 4WJI_1 Letter Amino acid
33 24 26 L Leucine
23 16 10 K Lycine
17 14 18 S Serine
15 27 23 V Valine
17 11 19 R Arginine
5 11 10 H Histidine
5 8 10 F Phenylalanine
12 18 12 P Proline
18 11 2 N Asparagine
12 12 8 Q Glutamine
24 17 21 I Isoleucine
9 8 5 M Methionine
16 14 20 T Threonine
7 4 4 W Tryptophan
2 3 2 C Cysteine
18 22 16 E Glutamic acid
14 35 26 G Glycine
9 7 3 Y Tyrosine
32 15 37 A Alanine
15 19 21 D Aspartic acid 

7MUC_1|Chains AA[auth BC], AB[auth CC], AC[auth DC], AD[auth EC], AE[auth FC], AF[auth GC], A[auth AC], NA[auth IC], NB[auth JC], NC[auth KC], ND[auth LC], NE[auth MC], N[auth HC]|DotC|Legionella pneumophila (446)
>8GST_1|Chains A, B, C, D|L-2,4-diketo-3-deoxyrhamnonate hydrolase|Sphingomonas sp. SKA58 (314266)
>4WJI_1|Chain A|Putative cyclohexadienyl dehydrogenase and ADH prephenate dehydrogenase|Rhizobium meliloti (266834)
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 \(w=f(c)\)
7MUC , Knot 136 303 0.85 40 192 295 
MRKFILSLSILLSALLVACSSRNHYGDTGSLAGLQAMADSKYTRAQKKQKMGKIREMALKETALSVGAQAGLAWRAKIIDEQLNKQARNLDAIYDFNSLVLEHNILPPVLLEGRNTLNLADAQSIRISDRTYKVAKQAHFITTPPTWRQYLWMDYVKPEAPNVTLLPKTKAEKEIWCIYTERGWKNGIDQANTILEENIARIKEDFGGMILYRKLLAMNMVSPPYVSHTDLGVTGDGSEIHIDDRVLRITALPELNVNSAEWRAAVAKDENALERFKNMEKLANQAKIVITNKSWQPIIAPVS
8GST , Knot 128 296 0.82 40 182 278 
MRGSHHHHHHGSKFCRFGQRGQEKPGIIDADGNIRDLSGVVPELTIDALAAAKGADIALLPLVEGEPRYGVPVKGIGKIVAIGLNYEDHAIESNLPIPTEPMMFMKALSSLNGPNDEVVLPKNSTHGDWEVELGVVIGETCRFVSEDEALSKVAGYVLVNDVSERFNQKQRGTQWSKGKGHDTFCPVGPWLVTPDEVGDPQDLDVHLDVNGERMQTGNTKTMIFNVAQLISYVSEYITLYPGDLMITGTPPGVGEGKKPQAIYLKAGDVMELGIEKLGTQRQQVSEWRHLGDEVFG
4WJI , Knot 125 293 0.80 40 174 279 
AQQFQTIALIGIGLIGSSIARDIREKQLAGTIVVTTRSEATLKRAGELGLGDRYTLSAAEAVEGADLVVVSVPVGASGAVAAEIAAHLKPGAIVTDVGSTKGSVIAQMAPHLPKDVHFVPGHPIAGTEHSGPDAGFAGLFRGRWCILTPPAGTDEEAVARLRLFWETLGSMVDEMDPKHHDKVLAIVSHLPHIIAYNIVGTADDLETVTESEVIKYSASGFRDFTRLAASDPTMWRDVCLHNKDAILEMLARFSEDLASLQRAIRWGDGDKLFDLFTRTRAIRRSIVQAGQDT

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 \(c\).

\(|P_{f(7MUC_1)}(2) \setminus P_{f(8GST_1)}(2)|=89\), \(|P_{f(8GST_1)}(2) \setminus P_{f(7MUC_1)}(2)|=79\). 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 1:100111010111011111000000010010111101110000001000001101001110001101110111110101100010001001011001001110001111111010001011010010100000011001011001101000111001010110101110001000110100001100110010011000110100011111100011110110110100001110101001010001101011101010010101111000011001001001100101110000101111110
Pair \(Z_2\) Length of longest common subsequence
7MUC_1,8GST_1 168 3
7MUC_1,4WJI_1 160 3
8GST_1,4WJI_1 164 3

Newick tree

 
[
	8GST_1:83.98,
	[
		7MUC_1:80,4WJI_1:80
	]:3.98
]

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 \((0.85)(0.8)(\frac{599 }{\log_{20} 599}-\frac{296}{\log_{20}296})=84.8\)
Status Protein1 Protein2 d d1/2
Query variables 7MUC_1 8GST_1 111 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]