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Parikh vectors
1QJH_1 8ECJ_1 4OXW_1 Letter Amino acid
0 3 4 H Histidine
5 13 7 P Proline
5 3 7 Y Tyrosine
7 13 2 N Asparagine
5 16 5 D Aspartic acid
0 0 0 C Cysteine
3 24 6 G Glycine
9 15 12 E Glutamic acid
6 15 6 I Isoleucine
1 5 3 W Tryptophan
11 38 9 V Valine
10 11 5 R Arginine
12 25 11 L Leucine
4 3 2 M Methionine
0 23 2 T Threonine
2 18 8 S Serine
9 50 6 A Alanine
6 12 6 Q Glutamine
4 11 15 K Lycine
2 9 3 F Phenylalanine

1QJH_1|Chain A|30S ribosomal protein S6|Thermus thermophilus (274)
>8ECJ_1|Chains A, B, C, D, E, F, G, H, I|Major capsid protein|Mycobacterium phage Cain (2015818)
>4OXW_1|Chain A|designed CISK-PX domain|Synthetic construct (32630)
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 \(w=f(c)\)
1QJH , Knot 53 101 0.80 34 84 98
MRRYEVNIVLNPNLDQSQLALEKEIIQRALENYGARVEKVAILGLMVLAYPIAKDPQGYFLWYQVEMPEDRVNDLARELRIRDNVRRVMVVKSQEPFLANA
8ECJ , Knot 129 307 0.80 38 166 278
MADISRAEVATLIQEAYSDTLLAAAKQGSTVLSAFQNVNMGTKTTHLPVLATLPEAGWVGESATEPEGVIPTSKVTWANRTLVAEEVAVIIPVPEAVIDDATVAILTEVAEQGGQAIGKKLDQAVIFGIDKPASWVSPALVPAAVAAGQAIAHVSGVANEFDLVGASNKVAEKVALAGWAPDTLLSSLALRYQVANVRDADGNLAFRDGSFLGFNTHFNRNGAWAPTSAVGVIADSSRVKIGVRQDITVKFLDQATLGTGENQINLAERDMVALRLKARFAYVLGVSATSVGANQTPVGVVTPDVTA
4OXW , Knot 62 119 0.83 38 97 114
SNAPDSLMKVSIPDFEKEGEGKSKHVMYKIKVKTGGEEWAVYRRYSDFYWLHKKLQQRYPELVPELPPKKWIYSALDEQILEKRKQGLEKYIQRIVSHPVLANDELVVSFLQAKAEHTG

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 \(c\).

\(|P_{f(1QJH_1)}(2) \setminus P_{f(8ECJ_1)}(2)|=43\), \(|P_{f(8ECJ_1)}(2) \setminus P_{f(1QJH_1)}(2)|=125\). 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 1:10000101110101000011100011001100011010011111111110111001010111001011000100110010100010011110000111101
Pair \(Z_2\) Length of longest common subsequence
1QJH_1,8ECJ_1 168 4
1QJH_1,4OXW_1 121 3
8ECJ_1,4OXW_1 161 4

Newick tree

 
[
	8ECJ_1:88.34,
	[
		1QJH_1:60.5,4OXW_1:60.5
	]:27.84
]

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 \((0.85)(0.8)(\frac{408 }{\log_{20} 408}-\frac{101}{\log_{20}101})=93.6\)
Status Protein1 Protein2 d d1/2
Query variables 1QJH_1 8ECJ_1 114 75.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]