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Parikh vectors
3KIN_1 4BBK_1 4FLT_1 Letter Amino acid
11 11 53 A Alanine
10 6 50 R Arginine
14 13 32 S Serine
10 7 29 F Phenylalanine
12 6 16 N Asparagine
4 4 4 C Cysteine
14 8 53 G Glycine
18 10 64 I Isoleucine
21 9 56 V Valine
15 9 39 D Aspartic acid
20 16 82 K Lycine
0 2 10 W Tryptophan
9 6 45 Y Tyrosine
6 6 11 M Methionine
10 8 38 P Proline
11 3 26 T Threonine
7 5 14 Q Glutamine
21 11 87 E Glutamic acid
9 7 19 H Histidine
16 18 65 L Leucine 

3KIN_1|Chains A, C|KINESIN HEAVY CHAIN|Rattus norvegicus (10116)
>4BBK_1|Chain A|FERMITIN FAMILY HOMOLOG 1|MUS MUSCULUS (10090)
>4FLT_1|Chain A|DNA polymerase 1|Pyrococcus abyssi (272844)
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 \(w=f(c)\)
3KIN , Knot 112 238 0.85 38 172 231 
ADPAECSIKVMCRFRPLNEAEILRGDKFIPKFKGEETVVIGQGKPYVFDRVLPPNTTQEQVYNACAKQIVKDVLEGYNGTIFAYGQTSSGKTHTMEGKLHDPQLMGIIPRIAHDIFDHIYSMDENLEFHIKVSYFEIYLDKIRDLLDVSKTNLAVHEDKNRVPYVKGCTERFVSSPEEVMDVIDEGKANRHVAVTNMNEHSSRSHSIFLINIKQENVETEKKLSGKLYLVDLAGSEKV
4BBK , Knot 75 165 0.77 40 124 160 
MAHHHHHHSSGLEVLFQGPEDITDIPKLADYLKLFRPKKLMLKACKQYWFVFKDTSIAYFKNKELEQGEPIEKLNLRGCEIVPDVNVSGRKFGIKLLIPVADGMNEVYLRCDHEDQYARWMAACILASKGKTMADSSYQPEVISILSFLKMKNRNSSPLVASSLE
4FLT , Knot 295 793 0.82 40 275 708 
MGSSHHHHHHSSGLVPAGSHAGMIIDADYITEDGKPIIRIFKKEKGEFKVEYDRTFRPYIYALLKDDSAIDEVKKITAERHGKIVRITEVEKVQKKFLGRPIEVWKLYLEHPQDVPAIREKIREHPAVVDIFEYDIPFAKRYLIDKGLTPMEGNEELTFLAVDIETLYHEGEEFGKGPIIMISYADEEGAKVITWKSIDLPYVEVVSSEREMIKRLVKVIREKDPDVIITYNGDNFAFPYLLKRAEKLGIKLPLGRDNSEPKMQRMGDSLAVEIKGRIHFDLFPVIRRTINLPTYTLEAVYEAIFGKSKEKVYAHEIAEAWETGKGLERVAKYSMEDAKVTFELGKEFFPMEAQLARLVGQPVWDVSRSSTGNLVEWFLLRKAYERNELAPNKPDEREYERRLRESYEGGYVKEPEKGLWEGIVSLDFRSLYPSIIITHNVSPDTLNRENCKEYDVAPQVGHRFCKDFPGFIPSLLGNLLEERQKIKKRMKESKDPVEKKLLDYRQRAIKILANSYYGYYGYAKARWYCKECAESVTAWGRQYIDLVRRELESRGFKVLYIDTDGLYATIPGAKHEEIKEKALKFVEYINSKLPGLLELEYEGFYARGFFVTKKKYALIDEEGKIVTRGLEIVRRDWSEIAKETQAKVLEAILKHGNVDEAVKIVKEVTEKLSKYEIPPEKLVIYEQITRPLSEYKAIGPHVAVAKRLAAKGVKVKPGMVIGYIVLRGDGPISKRAIAIEEFDPKKHKYDAEYYIENQVLPAVERILRAFGYRKEDLKYQKTKQVGLGAWLKF

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 \(c\).

\(|P_{f(3KIN_1)}(2) \setminus P_{f(4BBK_1)}(2)|=99\), \(|P_{f(4BBK_1)}(2) \setminus P_{f(3KIN_1)}(2)|=51\). 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 1:1011000101100101100101101001110101000111101010110011110000001001010011001101001011101000010000101010010111111011001100100100010101010010101001001101000011100000011010100001100100110110010100011100100000000011110100001000001010101101110001
Pair \(Z_2\) Length of longest common subsequence
3KIN_1,4BBK_1 150 3
3KIN_1,4FLT_1 171 4
4BBK_1,4FLT_1 199 10

Newick tree

 
[
	4FLT_1:97.97,
	[
		3KIN_1:75,4BBK_1:75
	]:22.97
]

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 \((0.85)(0.8)(\frac{403 }{\log_{20} 403}-\frac{165}{\log_{20}165})=71.0\)
Status Protein1 Protein2 d d1/2
Query variables 3KIN_1 4BBK_1 90 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]