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Parikh vectors
2IDZ_1 3AZR_1 6ZLY_1 Letter Amino acid
28 23 13 G Glycine
27 22 40 L Leucine
13 13 12 P Proline
17 15 15 V Valine
36 18 17 A Alanine
10 3 17 Q Glutamine
11 35 20 E Glutamic acid
9 4 10 M Methionine
16 14 21 S Serine
4 15 0 W Tryptophan
1 1 1 C Cysteine
5 10 13 H Histidine
20 29 23 I Isoleucine
7 18 15 F Phenylalanine
13 10 13 T Threonine
6 11 8 Y Tyrosine
13 18 9 R Arginine
8 15 9 N Asparagine
15 16 20 D Aspartic acid
9 27 24 K Lycine 

2IDZ_1|Chain A|Enoyl-[acyl-carrier-protein] reductase [NADH]|Mycobacterium tuberculosis (1773)
>3AZR_1|Chains A, B|Endoglucanase|Thermotoga maritima (243274)
>6ZLY_1|Chains A, B|Peroxisome proliferator-activated receptor gamma|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 \(w=f(c)\)
2IDZ , Knot 114 268 0.79 40 167 252 
TGLLDGKRILVSGIITDSSIAFHIARVAQEQGAQLVLTGFDRLRLIQRITDRLPAKAPLLELDVQNEEHLASLAGRVTEAIGAGNKLDGVVHSIGFMPQTGMGINPFFDAPYADVSKGIHISAYSYASMAKALLPIMNPGGSIVGMDFDPSRAMPAYNWMTVAKSALESVNRFVAREAGKYGVRSNLVAAGPIRTLAMSAIVGGALGEEAGAQIQLLEEGWDQRAPIGWNMKDATPVAKTVCALLSDWLPATTGDIIYADGGAHTQLL
3AZR , Knot 139 317 0.84 40 194 301 
MGVDPFERNKILGRGINIGNALEAPNEGDWGVVIKDEFFDIIKEAGFSHVRIPIRWSTHAYAFPPYKIMDRFFKRVDEVINGALKRGLAVVINIHHYEELMNDPEEHKERFLALWKQIADRYKDYPETLFFEILNEPHGNLTPEKWNELLEEALKVIRSIDKKHTIIIGTAEWGGISALEKLSVPKWEKNSIVTIHYYNPFEFTHQGAEWVEGSEKWLGRKWGSPDDQKHLIEEFNFIEEWSKKNKRPIYIGAFGAYRKADLESRIKWTSFVVREMEKRRWSWAYWEFCSGFGVYDTLRKTWNKDLLEALIGGDSIE
6ZLY , Knot 131 300 0.83 38 185 285 
HHHHHHSSGLVPRGSHMAEISSDIDQLNPESADLRALAKHLYDSYIKSFPLTKAKARAILTGKTTDKSPFVIYDMNSLMMGEDKIKFKHITPLQEQSKEVAIRIFQGCQFRSVEAVQEITEYAKSIPGFVNLDLNDQVTLLKYGVHEIIYTMLASLMNKDGVLISEGQGFMTREFLKSLRKPFGDFMEPKFEFAVKFNALELDDSDLAIFIAVIILSGDRPGLLNVKPIEDIQDNLLQALELQLKLNHPESSQLFAKLLQKMTDLRQIVTEHVQLLQVIKKTETDMSLHPLLQEIYKDLY

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 \(c\).

\(|P_{f(2IDZ_1)}(2) \setminus P_{f(3AZR_1)}(2)|=78\), \(|P_{f(3AZR_1)}(2) \setminus P_{f(2IDZ_1)}(2)|=105\). 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 1:0111010011101110000111011011000110111011001011001000111011110101000001101110100111110010111001111100111101110110101001101010001011011111101110111101010011110011011001100100111001100110001111111001110111111110011101011001100011111010010111001011100111100101101011100011
Pair \(Z_2\) Length of longest common subsequence
2IDZ_1,3AZR_1 183 4
2IDZ_1,6ZLY_1 158 4
3AZR_1,6ZLY_1 163 4

Newick tree

 
[
	3AZR_1:89.04,
	[
		2IDZ_1:79,6ZLY_1:79
	]:10.04
]

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 \((0.85)(0.8)(\frac{585 }{\log_{20} 585}-\frac{268}{\log_{20}268})=89.3\)
Status Protein1 Protein2 d d1/2
Query variables 2IDZ_1 3AZR_1 112 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]