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Parikh vectors
1GMW_1 1FEE_1 4RPT_1 Letter Amino acid
0 0 0 W Tryptophan
14 6 9 V Valine
3 2 6 N Asparagine
2 3 0 C Cysteine
3 8 11 K Lycine
5 1 11 F Phenylalanine
20 8 9 L Leucine
7 5 10 T Threonine
4 2 10 Y Tyrosine
13 3 3 A Alanine
10 1 5 R Arginine
10 5 6 E Glutamic acid
4 2 3 H Histidine
10 4 9 D Aspartic acid
11 6 7 G Glycine
6 5 13 S Serine
5 0 2 Q Glutamine
4 2 17 I Isoleucine
4 3 8 M Methionine
8 2 5 P Proline 

1GMW_1|Chains A, B, C|UREE|KLEBSIELLA AEROGENES (548)
>1FEE_1|Chains A, B|COPPER TRANSPORT PROTEIN ATOX1|Homo sapiens (9606)
>4RPT_1|Chains A, B|Capping enzyme protein|Rotavirus A (28875)
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 \(w=f(c)\)
1GMW , Knot 68 143 0.78 38 110 137 
MLYLTQRLEIPAAATASVTLPIDVRVKSRVKVTLNDGRDAGLLLPRGLLLRGGDVLSNEEGTEFVQVIAADEEVSVVRCDDPFMLAKACYALGNRHVPLQIMPGELRYHHDHVLDDMLRQFGLTVTFGQLPFEPEAGAYASES
1FEE , Knot 40 68 0.82 36 60 66 
MPKHEFSVDMTCGGCAEAVSRVLNKLGGVKYDIDLPNKKVCIESEHSMDTLLATLKKTGKTVSYLGLE
4RPT , Knot 73 144 0.84 36 112 138 
SGADDPNYFIGIKFRHIPYEYDVKIPHLTFGVLFISDNMIPDVVEIMKIMKKELFEMDITTSYTYMLSDGIYVANVSGVLATYFKMYNLFYKSQITFGQSRMFIPHITLSFSNNKTVRIESTRLKISSIYLRKIKGDTVFDMSE

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 \(c\).

\(|P_{f(1GMW_1)}(2) \setminus P_{f(1FEE_1)}(2)|=84\), \(|P_{f(1FEE_1)}(2) \setminus P_{f(1GMW_1)}(2)|=34\). 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 1:11010001011111010101110101000101010010011111101111011011000010011011110001011000011111010011100011101111010000001100110011101011011101011101000
Pair \(Z_2\) Length of longest common subsequence
1GMW_1,1FEE_1 118 3
1GMW_1,4RPT_1 146 4
1FEE_1,4RPT_1 122 3

Newick tree

 
[
	4RPT_1:69.80,
	[
		1GMW_1:59,1FEE_1:59
	]:10.80
]

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 \((0.85)(0.8)(\frac{211 }{\log_{20} 211}-\frac{68}{\log_{20}68})=47.4\)
Status Protein1 Protein2 d d1/2
Query variables 1GMW_1 1FEE_1 61 44.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]