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Parikh vectors
1KXU_1 2NNR_1 4YYH_1 Letter Amino acid
14 6 8 F Phenylalanine
20 9 9 A Alanine
8 10 5 G Glycine
14 6 4 H Histidine
14 1 6 I Isoleucine
11 3 7 M Methionine
15 7 6 P Proline
13 13 6 T Threonine
14 2 7 D Aspartic acid
8 0 1 C Cysteine
10 1 4 Q Glutamine
42 7 7 L Leucine
15 6 4 N Asparagine
31 8 5 E Glutamic acid
26 7 5 S Serine
2 2 0 W Tryptophan
20 2 3 R Arginine
24 7 12 K Lycine
15 4 5 Y Tyrosine
17 9 4 V Valine 

1KXU_1|Chain A|CYCLIN H|Homo sapiens (9606)
>2NNR_1|Chains A, B|Chagasin|Trypanosoma cruzi (5693)
>4YYH_1|Chains A, B|Bromodomain-containing protein 9|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 \(w=f(c)\)
1KXU , Knot 142 333 0.82 40 199 319 
MSHHHHHHGSMYHNSSQKRHWTFSSEEQLARLRADANRKFRCKAVANGKVLPNDPVFLEPHEEMTLCKYYEKRLLEFCSVFKPAMPRSVVGTACMYFKRFYLNNSVMEYHPRIIMLTCAFLACKVDEFNVSSPQFVGNLRESPLGQEKALEQILEYELLLIQQLNFHLIVHNPYRPFEGFLIDLKTRYPILENPEILRKTADDFLNRIALTDAYLLYTPSQIALTAILSSASRAGITMESYLSESLMLKENRTCLSQLLDIMKSMRNLVKKYEPPRSEEVAVLKQKLERCHSAELALNVITKKRKGYEDDDYVSKKSKHEEEEWTDDDLVESL
2NNR , Knot 55 110 0.78 38 87 106 
MSHKVTKAHNGATLTVAVGELVEIQLPSNPTTGFAWYFEGGTKESPNESMFTVENKYFPPDSKLLGAGGTEHFHVTVKAAGTHAVNLTYMRPWTGPSHDSERFTVYLKAN
4YYH , Knot 58 108 0.83 38 96 106 
GSAENESTPIQQLLEHFLRQLQRKDPHGFFAFPVTDAIAPGYSMIIKHPMDFGTMKDKIVANEYKSVTEFKADFKLMCDNAMTYNRPDTVYYKLAKKILHAGFKMMSK

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 \(c\).

\(|P_{f(1KXU_1)}(2) \setminus P_{f(2NNR_1)}(2)|=148\), \(|P_{f(2NNR_1)}(2) \setminus P_{f(1KXU_1)}(2)|=36\). 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 1:100000001010000000001010000011010101000100011101011100111101000101000000011010011011110011101010100101000110001011110011110010010100101110100011100011001100011110010101110010011011110100001110010110001001100111001011001001110111001001110100010001110000001001101100100110000110000111100010000010111011000001000000100000000001000011001
Pair \(Z_2\) Length of longest common subsequence
1KXU_1,2NNR_1 184 3
1KXU_1,4YYH_1 181 4
2NNR_1,4YYH_1 127 3

Newick tree

 
[
	1KXU_1:98.78,
	[
		4YYH_1:63.5,2NNR_1:63.5
	]:35.28
]

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 \((0.85)(0.8)(\frac{443 }{\log_{20} 443}-\frac{110}{\log_{20}110})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 1KXU_1 2NNR_1 127 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]