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Parikh vectors
1IHZ_1 3INT_1 4ARK_1 Letter Amino acid
6 20 10 N Asparagine
0 6 6 C Cysteine
12 25 24 E Glutamic acid
10 26 28 G Glycine
23 22 20 K Lycine
4 12 13 M Methionine
3 20 13 F Phenylalanine
5 17 16 R Arginine
4 20 15 H Histidine
7 28 9 Y Tyrosine
6 25 23 V Valine
14 21 19 A Alanine
8 22 18 D Aspartic acid
6 18 11 Q Glutamine
16 24 31 L Leucine
6 16 24 P Proline
5 19 27 S Serine
10 21 8 T Threonine
3 25 24 I Isoleucine
1 3 2 W Tryptophan 

1IHZ_1|Chain A|STAPHYLOCOCCAL NUCLEASE|Staphylococcus aureus (1280)
>3INT_1|Chains A, B|Probable UDP-galactopyranose mutase|Klebsiella pneumoniae (573)
>4ARK_1|Chain A|DUAL SPECIFICITY MITOGEN-ACTIVATED PROTEIN KINASE KINASE 1|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 \(w=f(c)\)
1IHZ , Knot 71 149 0.79 38 107 140 
ATSTKKLHKEPATLIKAIDGDTLKLMYKGQPMTFRLLLVDTPETKHPKKGVEKYGPEASAFTKKMLENAKKLEVEFDKGQRTDKYGRGLAYLYADGKMLNEALVRQGLAKVAYVYKPNNTHEQHLRKSEAQAKKEKLNIWSEDNADSGQ
3INT , Knot 167 390 0.85 40 234 372 
MKSKKILIVGAGFSGAVIGRQLAEKGHQVHIIDQRDHIGGNSYDARDSETNVMVHVYGPHIFHTDNETVWNYINKHAEMMPYVNRVKATVNGQVFSLPINLHTINQFFSKTCSPDEARALIAEKGDSTIADPQTFEEQALRFIGKELYEAFFKGYTIKQWGMQPSELPASILKRLPVRFNYDDNYFNHKFQGMPKCGYTQMIKSILNHENIKVDLQREFIVDERTHYDHVFYSGPLDAFYGYQYGRLGYRTLDFKKFIYQGDYQGCAVMNYCSVDVPYTRITEHKYFSPWEQHDGSVCYKEYSRACEENDIPYYPIRQMGEMALLEKYLSLAENETNITFVGRLGTYRYLDMDVTIAEALKTAEVYLNSLTDNQPMPVFTVSVGHHHHHH
4ARK , Knot 149 341 0.85 40 206 328 
MELKDDDFEKISELGAGNGGVVFKVSHKPSGLVMARKLIHLEIKPAIRNQIIRELQVLHECNSPYIVGFYGAFYSDGEISICMEHMDGGSLDQVLKKAGRIPEQILGKVSIAVIKGLTYLREKHKIMHRDVKPSNILVNSRGEIKLCDFGVSGQLIDSMANSFVGTRSYMSPERLQGTHYSVQSDIWSMGLSLVEMAVGRYPIPPPDAKELELMFGCQVEGDAAETPPRPRTPGRPLSSYGMDSRPPMAIFELLDYIVNEPPPKLPSGVFSLEFQDFVNKCLIKNPAERADLKQLMVHAFIKRSDAEEVDFAGWLCSTIGLNQPSTPTHAAGVLEHHHHHH

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 \(c\).

\(|P_{f(1IHZ_1)}(2) \setminus P_{f(3INT_1)}(2)|=38\), \(|P_{f(3INT_1)}(2) \setminus P_{f(1IHZ_1)}(2)|=165\). 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 1:10000010001101101101001011001011010111100100001001100011010110001100100101010010000001011101010101100111001110110100100000001000010100001011000010010
Pair \(Z_2\) Length of longest common subsequence
1IHZ_1,3INT_1 203 3
1IHZ_1,4ARK_1 187 3
3INT_1,4ARK_1 158 6

Newick tree

 
[
	1IHZ_1:10.03,
	[
		4ARK_1:79,3INT_1:79
	]:24.03
]

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 \((0.85)(0.8)(\frac{539 }{\log_{20} 539}-\frac{149}{\log_{20}149})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 1IHZ_1 3INT_1 144 99
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]