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Parikh vectors
5ZUW_1 9HMY_1 5QLP_1 Letter Amino acid
9 2 6 M Methionine
25 35 21 V Valine
14 26 3 R Arginine
22 7 7 N Asparagine
14 30 12 D Aspartic acid
2 2 2 C Cysteine
19 6 14 Q Glutamine
50 35 17 A Alanine
46 37 12 L Leucine
13 17 9 T Threonine
8 6 8 Y Tyrosine
23 31 13 G Glycine
7 7 3 H Histidine
20 12 17 K Lycine
19 8 11 F Phenylalanine
26 17 7 P Proline
29 17 12 E Glutamic acid
17 21 5 I Isoleucine
27 13 8 S Serine
3 0 2 W Tryptophan 

5ZUW_1|Chain A|Mannose-6-phosphate isomerase|Salmonella typhimurium (strain LT2 / SGSC1412 / ATCC 700720) (99287)
>9HMY_1|Chain A|SteA-like C-terminal domain-containing protein|Mycobacterium tuberculosis H37Rv (83332)
>5QLP_1|Chains A, B|Thiol:disulfide interchange protein|Escherichia coli K-12 (83333)
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 \(w=f(c)\)
5ZUW , Knot 165 393 0.83 40 210 366 
ASMQKLINSVQNYAWGSKTALTELYGIANPQQQPMAELWMGAHPKSSSRITTANGETVSLRDAIEKNKTAMLGEAVANRFGELPFLFKVLCAAQPLSIQVQPNKRNSEIGFAKENAAGIPMDAAERNYKDPNHKPELVFALTPFLAMNAFREFSDIVSLLQPVAGAHSAIAHFLQVPNAERLSQLFASLLNMQGEEKSRALAVLKAALNSQQGEPWQTIRVISEYYPDDSGLFSPLLLNVVKLNPGEAMFLFAETPHAYLQGVALEVMANSDNVLRAGLTPKYIDIPELVANVKFEPKPAGELLTAPVKSGAELDFPIPVDDFAFSLHDLALQETSIGQHSAAILFCVEGEAVLRKDEQRLVLKPGESAFIGADESPVNASGTGRLARVYNKL
9HMY , Knot 137 329 0.80 38 176 311 
RPGLIGIARVDRNIDRLLRRVCPGDIVVLDVLDLDRITADALVEAEIAAVVNASSSVSGRYPNLGPEVLVTNGVTLIDETGPEIFKKVKDGAKVRLYEGGVYAGDRRLIRGTERTDHDIADLMREAKSGLVAHLEAFAGNTIEFIRSESPLLIDGIGIPDVDVDLRRRHVVIVADEPSGPDDLKSLKPFIKEYQPVLVGVGTGADVLRKAGYRPQLIVGDPDQISTEVLKCGAQVVLPADADGHAPGLERIQDLGVGAMTFPAAGSATDLALLLADHHGAALLVTAGHAANIETFFDRTRVQSNPSTFLTRLRVGEKLVDAKAVATLYR
5QLP , Knot 86 189 0.79 40 134 185 
AQYEDGKQYTTLEKPVAGAPQVLEFFSFFCPHCYQFEEVLHISDNVKKKLPEGVKMTKYHVNFMGGDLGKDLTQAWAVAMALGVEDKVTVPLFEGVQKTQTIRSASDIRDVFINAGIKGEEYDAAWNSFVVKSLVAQQEKAAADVQLRGVPAMFVNGKYQLNPQGMDTSNMDVFVQQYADTVKYLSEKK

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 \(c\).

\(|P_{f(5ZUW_1)}(2) \setminus P_{f(9HMY_1)}(2)|=89\), \(|P_{f(9HMY_1)}(2) \setminus P_{f(5ZUW_1)}(2)|=55\). 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 1:101001100100011100011001011101000111011111010000010010100101001100000111101110011011111011011011010101000000111100011111101100000010001011111011111011001001101101111100111011011010010011101101010000011111011100001011001011000010001110111101101011011111100101010111101110000110111010010110111010101011101101110011010111110011101001110000110001111101010111000000111011001111100011010101011010001
Pair \(Z_2\) Length of longest common subsequence
5ZUW_1,9HMY_1 144 4
5ZUW_1,5QLP_1 160 5
9HMY_1,5QLP_1 160 3

Newick tree

 
[
	5QLP_1:82.49,
	[
		5ZUW_1:72,9HMY_1:72
	]:10.49
]

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 \((0.85)(0.8)(\frac{722 }{\log_{20} 722}-\frac{329}{\log_{20}329})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 5ZUW_1 9HMY_1 132 122.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]