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Parikh vectors
1NKZ_1 6MVL_1 4MRI_1 Letter Amino acid
0 6 9 C Cysteine
3 6 19 P Proline
6 12 15 V Valine
7 10 18 A Alanine
0 10 12 R Arginine
0 8 19 D Aspartic acid
1 6 13 Y Tyrosine
0 8 23 E Glutamic acid
1 4 13 F Phenylalanine
3 1 2 W Tryptophan
2 20 14 H Histidine
6 4 27 I Isoleucine
5 15 28 L Leucine
2 4 15 N Asparagine
2 11 9 Q Glutamine
5 11 18 G Glycine
4 12 18 T Threonine
3 4 23 K Lycine
1 2 7 M Methionine
2 15 11 S Serine 

1NKZ_1|Chains A, C, E|Light-harvesting protein B-800/850, alpha chain|Rhodoblastus acidophilus (1074)
>6MVL_1|Chain A|V-type immunoglobulin domain-containing suppressor of T-cell activation|Homo sapiens (9606)
>4MRI_1|Chains A, B|Aspartoacylase|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 \(w=f(c)\)
1NKZ , Knot 32 53 0.80 32 45 50 
MNQGKIWTVVNPAIGIPALLGSVTVIAILVHLAILSHTTWFPAYWQGGVKKAA
6MVL , Knot 83 169 0.84 40 128 161 
AFKVATPYSLYVCPEGQNVTLTCRLLGPVDKGHDVTFYKTWYRSSRGEVQTCSERRPIRQLTFQDLHLHHGGHQAAQTSHDLAQRHGLESASDHHGNFSITMRNLTLLDSGLYCCLVVEIRHHHSEHRVHGAMELQVQTGKDAPSNCVVYPSSSQDSEQITAAHHHHHH
4MRI , Knot 136 313 0.83 40 196 302 
MTSCHIAEEHIQKVAIFGGTHGNELTGVFLVKHWLENGAEIQRTGLEVKPFITNPRAVKKCTRYIDCDLNRIFDLENLGKKMSEDLPYEVRRAQEINHLFGPKDSEDSYDIIFDLHNTTSNMGCTLILEDSRNNFLIQMFHYIKTSLAPLPCYVYLIEHPSLKYATTRSIAKYPVGIEVGPQPQGVLRADILDQMRKMIKHALDFIHHFNEGKEFPPCAIEVYKIIEKVDYPRDENGEIAAIIHPNLQDQDWKPLHPGDPMFLTLDGKTIPLGGDCTVYPVFVNEAAYYEKKEASAKTTKLTLNAKSIRCCLH

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 \(c\).

\(|P_{f(1NKZ_1)}(2) \setminus P_{f(6MVL_1)}(2)|=32\), \(|P_{f(6MVL_1)}(2) \setminus P_{f(1NKZ_1)}(2)|=115\). 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 1:10010110110111111111101011111101111000011110101110011
Pair \(Z_2\) Length of longest common subsequence
1NKZ_1,6MVL_1 147 3
1NKZ_1,4MRI_1 193 3
6MVL_1,4MRI_1 198 3

Newick tree

 
[
	4MRI_1:10.60,
	[
		1NKZ_1:73.5,6MVL_1:73.5
	]:31.10
]

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 \((0.85)(0.8)(\frac{222 }{\log_{20} 222}-\frac{53}{\log_{20}53})=56.5\)
Status Protein1 Protein2 d d1/2
Query variables 1NKZ_1 6MVL_1 72 46.5
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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