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Parikh vectors
2XYP_1 1FRF_1 7GXC_1 Letter Amino acid
3 8 3 H Histidine
11 17 4 K Lycine
3 22 3 P Proline
3 24 8 A Alanine
1 7 4 Q Glutamine
11 19 6 E Glutamic acid
11 15 9 I Isoleucine
12 19 15 L Leucine
4 4 6 M Methionine
0 4 0 W Tryptophan
7 9 7 F Phenylalanine
12 10 9 S Serine
10 15 7 T Threonine
3 9 3 Y Tyrosine
7 17 10 V Valine
11 4 9 R Arginine
10 11 7 N Asparagine
10 11 8 D Aspartic acid
5 12 5 C Cysteine
12 27 5 G Glycine 

2XYP_1|Chain A|CASPASE-3 SUBUNIT P17|HOMO SAPIENS (9606)
>1FRF_1|Chain A[auth S]|[NI-FE] HYDROGENASE|Desulfovibrio fructosovorans (878)
>7GXC_1|Chain A|B-cell lymphoma 6 protein|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 \(w=f(c)\)
2XYP , Knot 72 146 0.82 38 113 143 
SGISLDNSYKMDYPEMGLCIIINNKNFHKSTGMTSRSGTDVDAANLRETFRNLKYEVRNKNDLTREEIVELMRDVSKEDHSKRSSFVCVLLSHGEEGIIFGTNGPVDLKKITNFFRGDRCRSLTGKPKLFIIQACRGTELDCGIET
1FRF , Knot 118 264 0.83 40 172 250 
LTAKHRPSVVWLHNAECTGCTEAAIRTIKPYIDALILDTISLDYQETIMAAAGETSEAALHEALEGKDGYYLVVEGGLPTIDGGQWGMVAGHPMIETCKKAAAKAKGIICIGTCSPYGGVQKAKPNPSQAKGVSEALGVKTINIPGCPPNPINFVGAVVHVLTKGIPDLDENGRPKLFYGELVHDNCPRLPHFEASEFAPSFDSEEAKKGFCLYELGCKGPVTYNNCPKVLFNQVNWPVQAGHPCLGCSEPDFWDTMTPFYEQG
7GXC , Knot 65 128 0.82 38 98 125 
GPGADSCIQFTRHASDVLLNLNRLRSRDILTDVVIVVSREQFRAHKTVLMACSGLFYSIFTDQLKCNLSVINLDPEINPEGFCILLDFMYTSRLNLREGNIMAVMATAMYLQMEHVVDTCRKFIKASE

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 \(c\).

\(|P_{f(2XYP_1)}(2) \setminus P_{f(1FRF_1)}(2)|=69\), \(|P_{f(1FRF_1)}(2) \setminus P_{f(2XYP_1)}(2)|=128\). 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 1:01101000001001011101110000100001100001001011010001001000100000100001101100100000000001101110010011111001110100100110100000101010111101001001001100
Pair \(Z_2\) Length of longest common subsequence
2XYP_1,1FRF_1 197 4
2XYP_1,7GXC_1 139 4
1FRF_1,7GXC_1 192 3

Newick tree

 
[
	1FRF_1:10.89,
	[
		2XYP_1:69.5,7GXC_1:69.5
	]:35.39
]

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 \((0.85)(0.8)(\frac{410 }{\log_{20} 410}-\frac{146}{\log_{20}146})=79.1\)
Status Protein1 Protein2 d d1/2
Query variables 2XYP_1 1FRF_1 101 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]