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Parikh vectors
9CSF_1 5QAO_1 1JDV_1 Letter Amino acid
5 12 8 R Arginine
9 18 16 K Lycine
2 7 4 M Methionine
0 10 3 W Tryptophan
3 6 11 Y Tyrosine
0 0 2 C Cysteine
4 15 22 G Glycine
7 17 12 I Isoleucine
10 17 25 L Leucine
3 11 9 F Phenylalanine
4 6 6 P Proline
5 14 16 S Serine
7 12 17 T Threonine
7 7 2 H Histidine
6 16 23 V Valine
4 16 19 A Alanine
2 16 14 N Asparagine
9 14 8 D Aspartic acid
7 14 3 Q Glutamine
7 15 16 E Glutamic acid 

9CSF_1|Chains A, B|Ubiquitin Variant UbV3|synthetic construct (32630)
>5QAO_1|Chains A, B, C, D|Beta-lactamase|Klebsiella pneumoniae (573)
>1JDV_1|Chains A, B, C, D, E, F|5'-METHYLTHIOADENOSINE PHOSPHORYLASE|Sulfolobus solfataricus (2287)
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 \(w=f(c)\)
9CSF , Knot 51 101 0.77 36 79 95 
MAHHHHHHVTSLYKKAGSTDYKDDDDKMQILVRTPFVQFITLEVEPSDTIENVKAKIQEKEGIPPDQQRLIFAGKQLEDGRTLSDYNIQKESTLHLVLRLR
5QAO , Knot 112 243 0.84 38 166 239 
KEWQENKSWNAHFTEHKSQGVVVLWNENKQQGFTNNLKRANQAFLPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIATWNRDHNLITAMKYSVVPVYQEFARQIGEARMSKMLHAFDYGNEDISGNVDSFWLDGGIRISATEQISFLRKLYHNKLHVSERSQRIVKQAMLTEANGDYIIRAKTGYSTRIEPKIGWWVGWVELDDNVWFFAMNMDMPTSDGLGLRQAITKEVLKQEKIIP
1JDV , Knot 108 236 0.83 40 160 226 
MNPVHILAKKGEVAERVLVVGDPGRARLLSTLLQNPKLTNENRGFLVYTGKYNGETVSIATHGIGGPSIAIVLEELAMLGANVFIRYGTTGALVPYINLGEYIIVTGASYNQGGLFYQYLRDNACVASTPDFELTNKLVTSFSKRNLKYYVGNVFSSDAFYAEDEEFVKKWSSRGNIAVEMECATLFTLSKVKGWKSATVLVVSDNLAKGGIWITKEELEKSVMDGAKAVLDTLTS

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 \(c\).

\(|P_{f(9CSF_1)}(2) \setminus P_{f(5QAO_1)}(2)|=42\), \(|P_{f(5QAO_1)}(2) \setminus P_{f(9CSF_1)}(2)|=129\). 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 1:11000000100100011000000000010111001110110101010001001010100001111000011111001001001000010000010111010
Pair \(Z_2\) Length of longest common subsequence
9CSF_1,5QAO_1 171 3
9CSF_1,1JDV_1 167 3
5QAO_1,1JDV_1 158 4

Newick tree

 
[
	9CSF_1:86.26,
	[
		1JDV_1:79,5QAO_1:79
	]:7.26
]

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 \((0.85)(0.8)(\frac{344 }{\log_{20} 344}-\frac{101}{\log_{20}101})=75.3\)
Status Protein1 Protein2 d d1/2
Query variables 9CSF_1 5QAO_1 99 69
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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