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Parikh vectors
8ZWR_1 5HNI_1 6FJB_1 Letter Amino acid
17 13 19 R Arginine
8 14 19 N Asparagine
16 15 21 E Glutamic acid
19 30 14 V Valine
5 4 3 W Tryptophan
9 15 24 A Alanine
15 17 24 D Aspartic acid
10 16 28 I Isoleucine
18 19 16 S Serine
6 9 9 M Methionine
14 20 16 G Glycine
5 14 19 H Histidine
29 34 44 L Leucine
21 21 23 K Lycine
18 16 14 T Threonine
10 9 19 Y Tyrosine
7 6 7 C Cysteine
13 8 12 Q Glutamine
24 17 13 F Phenylalanine
14 15 20 P Proline 

8ZWR_1|Chains A, B|Beta-1,3-galactosyltransferase 5|Homo sapiens (9606)
>5HNI_1|Chains A[auth X], B[auth Y]|Hepatocyte growth factor receptor|Homo sapiens (9606)
>6FJB_1|Chain A|Mitogen-activated protein kinase 1|Rattus norvegicus (10116)
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 \(w=f(c)\)
8ZWR , Knot 121 278 0.81 40 183 264 
FKEQSFVYKKDGNFLKLPDTDCRQTPPFLVLLVTSSHKQLAERMAIRQTWGKERMVKGKQLKTFFLLGTTSSAAETKEVDQESQRHGDIIQKDFLDVYYNLTLKTMMGIEWVHRFCPQAAFVMKTDSDMFINVDYLTELLLKKNRTTRFFTGFLKLNEFPIRQPFSKWFVSKSEYPWDRYPPFCSGTGYVFSGDVASQVYNVSKSVPYIKLEDVFVGLCLERLNIRLEELHSQPTFFPGGLRFSVCLFRRIVACHFIKPRTLLDYWQALENSRGEDCP
5HNI , Knot 140 312 0.86 40 203 303 
NTVHIDLSALNPELVQAVQHVVIGPSSLIVHFNEVIGRGHFGCVYHGTLLDNDGKKIHCAVKSLNRITDIGEVSQFLTEGIIMKDFSHPNVLSLLGICLRSEGSPLVVLPYMKHGDLRNFIRNETHNPTVKDLIGFGLQVAKGMKFLASKKFVHRDLAARNCMLDEKFTVKVADFGLARDMYDKEFDSVHNKTGAKLPVKWMALESLQTQKFTTKSDVWSFGVLLWELMTRGAPPYPDVNTFDITVYLLQGRRLLQPEYCPDPLYEVMLKCWHPKAEMRPSFSELVSRISAIFSTFIGEHYVHVNATYVNVK
6FJB , Knot 157 364 0.84 40 235 349 
HHHHHHMAAAAAAGPEMVRGQVFDVGPRYTNLSYIGEGAYGMVCSAYDNLNKVRVAIKKISPFEHQTYCQRTLREIKILLRFRHENIIGINDIIRAPTIEQMKDVYIVQDLMETDLYKLLKTQHLSNDHICYFLYQILRGLKYIHSANVLHRDLKPSNLLLNTTCDLKICDFGLARVADPDHDHTGFLTEYVATRWYRAPEIMLNSKGYTKSIDIWSVGCILAEMLSNRPIFPGKHYLDQLNHILGILGSPSQEDLNCIINLKARNYLLSLPHKNKVPWNRLFPNADSKALDLLDKMLTFNPHKRIEVEQALAHPYLEQYYDPSDEPIAEAPFKFDMELDDLPKEKLKELIFEETARFQPGYRS

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 \(c\).

\(|P_{f(8ZWR_1)}(2) \setminus P_{f(5HNI_1)}(2)|=80\), \(|P_{f(5HNI_1)}(2) \setminus P_{f(8ZWR_1)}(2)|=100\). 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 1:10000110000101101100000001111111100000011001110001100011010010011111000011000010000000101100011010001010011110110010101111100000111010010011100000001101110100111001100111000001100011100101011010110010010001101010011111010010101001000101111110101011001110011010011001011000010001
Pair \(Z_2\) Length of longest common subsequence
8ZWR_1,5HNI_1 180 4
8ZWR_1,6FJB_1 190 4
5HNI_1,6FJB_1 166 6

Newick tree

 
[
	8ZWR_1:95.50,
	[
		5HNI_1:83,6FJB_1:83
	]:12.50
]

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 \((0.85)(0.8)(\frac{590 }{\log_{20} 590}-\frac{278}{\log_{20}278})=87.7\)
Status Protein1 Protein2 d d1/2
Query variables 8ZWR_1 5HNI_1 115 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]