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Parikh vectors
1YUE_1 1ESK_1 7LPM_1 Letter Amino acid
42 2 12 A Alanine
30 1 7 D Aspartic acid
15 1 2 P Proline
19 3 14 N Asparagine
32 3 2 E Glutamic acid
3 2 1 H Histidine
28 7 6 K Lycine
15 1 3 F Phenylalanine
31 0 10 S Serine
19 0 3 Y Tyrosine
28 1 6 V Valine
15 4 11 R Arginine
11 2 3 Q Glutamine
30 0 6 I Isoleucine
44 0 8 L Leucine
30 2 7 T Threonine
3 6 8 C Cysteine
24 5 12 G Glycine
6 1 2 M Methionine
2 1 6 W Tryptophan 

1YUE_1|Chain A|Head vertex protein Gp24|Enterobacteria phage T4 (10665)
>1ESK_1|Chain A|GAG POLYPROTEIN|null
>7LPM_1|Chain A|Lysozyme C|Gallus gallus (9031)
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 \(w=f(c)\)
1YUE , Knot 177 427 0.83 40 214 397 
MAKINELLRESTTTNSNSIGRPNLVALTRATTKLIYSDIVATQRTNQPVAAFYGIKYLNPDNEFTFKTGATYAGEAGYVDREQITELTEESKLTLNKGDLFKYNNIVYKVLEDTPFATIEESDLELALQIAIVLLKVRLFSDAASTSKFESSDSEIADARFQINKWQTAVKSRKLKTGITVELAQDLEANGFDAPNFLEDLLATEMADEINKDILQSLITVSKRYKVTGITDSGFIDLSYASAPEAGRSLYRMVCEMVSHIQKESTYTATFCVASARAAAILAASGWLKHKPEDDKYLSQNAYGFLANGLPLYCDTNSPLDYVIVGVVENIGEKEIVGSIFYAPYTEGLDLDDPEHVGAFKVVVDPESLQPSIGLLVRYALSANPYTVAKDEKEARIIDGGDMDKMAGRSDLSVLLGVKLPKIIIDE
1ESK , Knot 24 42 0.71 32 34 36 
NVKCFNCGKEGHTARNCRAPRKKGCWKCGKEGHQMKDCTERQ
7LPM , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL

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 \(c\).

\(|P_{f(1YUE_1)}(2) \setminus P_{f(1ESK_1)}(2)|=203\), \(|P_{f(1ESK_1)}(2) \setminus P_{f(1YUE_1)}(2)|=23\). 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 1:1101001100000000001101011110010001100011100000011111011001010001010011001101101000010010000010100101100001100110001110100001011101111110101100110000100000011010101001001100001001101011001010110110110011100110010001100110100000101100011101001011011001001100110010000000101011010111111101110001000001000101111011110000001100111111001100011101101100011010010011110111010010101111100110101001100000101101101001110001011111011011100
Pair \(Z_2\) Length of longest common subsequence
1YUE_1,1ESK_1 226 2
1YUE_1,7LPM_1 196 4
1ESK_1,7LPM_1 120 2

Newick tree

 
[
	1YUE_1:11.11,
	[
		7LPM_1:60,1ESK_1:60
	]:57.11
]

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 \((0.85)(0.8)(\frac{469 }{\log_{20} 469}-\frac{42}{\log_{20}42})=132.\)
Status Protein1 Protein2 d d1/2
Query variables 1YUE_1 1ESK_1 169 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]