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Parikh vectors
1AHT_1 7VHZ_1 6TUL_1 Letter Amino acid
4 26 23 G Glycine
5 36 22 L Leucine
1 18 10 Y Tyrosine
0 11 7 N Asparagine
1 19 5 C Cysteine
0 18 6 Q Glutamine
1 29 13 P Proline
0 25 21 V Valine
1 20 3 I Isoleucine
3 33 12 K Lycine
2 27 6 F Phenylalanine
1 21 21 A Alanine
3 28 10 R Arginine
0 16 4 H Histidine
3 29 32 S Serine
2 22 17 T Threonine
0 5 4 W Tryptophan
3 28 8 D Aspartic acid
6 34 7 E Glutamic acid
0 9 2 M Methionine 

1AHT_1|Chain A[auth L]|ALPHA-THROMBIN (SMALL SUBUNIT)|Homo sapiens (9606)
>7VHZ_1|Chains A, B|Histone acetyltransferase p300|Homo sapiens (9606)
>6TUL_1|Chain A[auth HHH]|Fab C0021177 heavy chain (IgG1)|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 \(w=f(c)\)
1AHT , Knot 24 36 0.79 28 34 34 
TFGSGEADCGLRPLFEKKSLEDKTERELLESYIDGR
7VHZ , Knot 194 454 0.87 40 259 435 
GPSLGYCCGRKLEFSPQTLCCYGKQLCTIPRDATYYSYQNRYHFCEKCFNEIQGESVSLGDDPSQPQTTINKEQFSKRKNDTLDPELFVECTECGRKMHQICVLHHEIIWPAGFVCDGCLKKSARTRKENKFSAKRLPSTRLGTFLENRVNDFLRRQNHPESGEVTVRVVHASDKTVEVKPGMKARFVDSGEMAESFPYRTKALFAFEEIDGVDLCFFGMHVQEYGSDCPPPNQRRVYISYLDSVHFFRPKCLRTAVYHEILIGYLEYVKKLGYTTGHIWACPPSEGDDYIFHCHPPDQKIPKPKRLQEWFKKMLDKAVSERIVHDYKDIFKQATEDRLTSAKELPYFEGDFWPNVLEESIKESGGSGSQKLYATMEKHKEVFFVIRLIAGPAANSLPPIVDPDPLIPCDLMDGRDAFLTLARDKHLEFSSLRRAQWSTMCMLVELHTQSQDRF
6TUL , Knot 102 233 0.79 40 142 222 
GAHSEVQLLESGGGLVQPGGSLRLSCAASGFTFRYEVAAWVRQAPGKGLEWVSAISGPAPKGYYADSVKGRFTISRDNSKNTLYLQMNSLRAEDTAVYYCARLRADLGLYMDLWGRGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKRVEPKSCDKTHAA

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 \(c\).

\(|P_{f(1AHT_1)}(2) \setminus P_{f(7VHZ_1)}(2)|=5\), \(|P_{f(7VHZ_1)}(2) \setminus P_{f(1AHT_1)}(2)|=230\). 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 1:011010100110111000010000000110001010
Pair \(Z_2\) Length of longest common subsequence
1AHT_1,7VHZ_1 235 3
1AHT_1,6TUL_1 142 4
7VHZ_1,6TUL_1 187 4

Newick tree

 
[
	7VHZ_1:11.55,
	[
		1AHT_1:71,6TUL_1:71
	]:44.55
]

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 \((0.85)(0.8)(\frac{490 }{\log_{20} 490}-\frac{36}{\log_{20}36})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 1AHT_1 7VHZ_1 180 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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