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Parikh vectors
8ATT_1 5THO_1 7RWA_1 Letter Amino acid
18 19 41 E Glutamic acid
18 13 30 T Threonine
12 21 33 R Arginine
20 21 26 G Glycine
12 3 11 M Methionine
25 16 49 S Serine
16 10 22 Y Tyrosine
12 35 54 A Alanine
14 7 29 N Asparagine
9 10 23 Q Glutamine
24 9 40 I Isoleucine
19 8 23 P Proline
9 0 4 W Tryptophan
16 17 50 V Valine
27 13 30 D Aspartic acid
5 1 11 C Cysteine
12 2 14 H Histidine
36 20 80 L Leucine
30 7 39 K Lycine
20 8 21 F Phenylalanine 

8ATT_1|Chain A[auth B]|Mitochondrial transcription factor 1|Saccharomyces cerevisiae S288C (559292)
>5THO_1|Chains A, B, C, D, E, F, G, O, P, Q, R, S, T, U|Proteasome subunit alpha|Mycobacterium tuberculosis (strain ATCC 25177 / H37Ra) (419947)
>7RWA_1|Chains A, F[auth a]|AP-2 complex subunit alpha-2|Mus musculus (10090)
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 \(w=f(c)\)
8ATT , Knot 151 354 0.83 40 220 336 
MGGSHHHHHHGMASSVPIPGIKDISKLKFFYGFKYLWNPTVYNKIFDKLDLTKTYKHPEELKVLDLYPGVGIQSAIFYNKYCPRQYSLLEKRSSLYKFLNAKFEGSPLQILKRDPYDWSTYSNLIDEERIFVPEVQSSDHINDKFLTVANVTGEGSEGLIMQWLSCIGNKNWLYRFGKVKMLLWMPSTTARKLLARPGMHSRSKCSVVREAFTDTKLIAISDANELKGFDSQCIEEWDPILFSAAEIWPTKGKPIALVEMDPIDFDFDVDNWDYVTRHLMILKRTPLNTVMDSLGHGGQQYFNSRITDKDLLKKCPIDLTNDEFIYLTKLFMEWPFKPDILMDFVDMYQTEHSG
5THO , Knot 107 240 0.81 38 148 225 
MEQAMRERSELARKGIARAKSVVALAYAGGVLFVAENPSRSLQKISELYDRVGFAAAGKFNEFDNLRRGGIQFADTRGYAYDRRDVTGRQLANVYAQTLGTIFTEQAKPYEVELCVAEVAHYGETKRPELYRITYDGSIADEPHFVVMGGTTEPIANALKESYAENASLTDALRIAVAALRAGSADTSGGDQPTLGVASLEVAVLDANRPRRAFRRITGSALQALLVDQESPQSDGESSG
7RWA , Knot 246 630 0.84 40 256 577 
MPAVSKGDGMRGLAVFISDIRNCKSKEAEIKRINKELANIRSKFKGDKALDGYSKKKYVCKLLFIFLLGHDIDFGHMEAVNLLSSNRYTEKQIGYLFISVLVNSNSELIRLINNAIKNDLASRNPTFMGLALHCIANVGSREMAEAFAGEIPKILVAGDTMDSVKQSAALCLLRLYRTSPDLVPMGDWTSRVVHLLNDQHLGVVTAATSLITTLAQKNPEEFKTSVSLAVSRLSRIVTSASTDLQDYTYYFVPAPWLSVKLLRLLQCYPPPEDPAVRGRLTECLETILNKAQEPPKSKKVQHSNAKNAVLFEAISLIIHHDSEPNLLVRACNQLGQFLQHRETNLRYLALESMCTLASSEFSHEAVKTHIETVINALKTERDVSVRQRAVDLLYAMCDRSNAQQIVAEMLSYLETADYSIREEIVLKVAILAEKYAVDYTWYVDTILNLIRIAGDYVSEEVWYRVIQIVINRDDVQGYAAKTVFEALQAPACHENLVKVGGYILGEFGNLIAGDPRSSPLIQFNLLHSKFHLCSVPTRALLLSTYIKFVNLFPEVKATIQDVLRSDSQLKNADVELQQRAVEYLRLSTVASTDILATVLEEMPPFPERESSILAKLKKKKGGSGLEVLFQ

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 \(c\).

\(|P_{f(8ATT_1)}(2) \setminus P_{f(5THO_1)}(2)|=121\), \(|P_{f(5THO_1)}(2) \setminus P_{f(8ATT_1)}(2)|=49\). 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 1:111000000011100111111001001011011001101010001100101000000100101101011111001110000010000110000010011010101011011000100100000110000111101000001000110110101010011110110011000110011010111111000100111011100000001100110000111100100101100001001011110110111001011111010110101010010010001111000110011001101100010001000011000110100001101001110111010111011010000001
Pair \(Z_2\) Length of longest common subsequence
8ATT_1,5THO_1 170 3
8ATT_1,7RWA_1 164 3
5THO_1,7RWA_1 180 4

Newick tree

 
[
	5THO_1:89.30,
	[
		8ATT_1:82,7RWA_1:82
	]:7.30
]

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 \((0.85)(0.8)(\frac{594 }{\log_{20} 594}-\frac{240}{\log_{20}240})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 8ATT_1 5THO_1 128 106.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]