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Parikh vectors
8FAJ_1 3AHC_1 2VWQ_1 Letter Amino acid
16 46 7 N Asparagine
0 6 1 C Cysteine
15 59 35 E Glutamic acid
21 69 30 A Alanine
14 59 29 D Aspartic acid
20 62 27 L Leucine
9 21 7 M Methionine
12 40 18 F Phenylalanine
12 50 20 T Threonine
13 34 17 R Arginine
14 28 6 Q Glutamine
16 64 30 G Glycine
16 45 18 S Serine
7 35 12 H Histidine
19 43 16 I Isoleucine
18 41 12 K Lycine
7 44 24 P Proline
10 17 3 W Tryptophan
6 31 10 Y Tyrosine
20 51 35 V Valine 

8FAJ_1|Chains A, B|Beta-lactamase|Klebsiella pneumoniae (573)
>3AHC_1|Chain A|Xylulose 5-phosphate/fructose 6-phosphate phosphoketolase|Bifidobacterium breve (1685)
>2VWQ_1|Chain A|GLUCOSE DEHYDROGENASE|HALOFERAX MEDITERRANEI (2252)
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 \(w=f(c)\)
8FAJ , Knot 121 265 0.85 38 175 261 
MRVLALSAVFLVASIIGMPAVAKEWQENKSWNAHFTEHKSQGVVVLWNENKQQGFTNNLKRANQAFLPASTFKIPNSLIALDLGVVKDEHQVFKWDGQTRDIATWNRDHNLITAMKYSVVPVYQEFARQIGEARMSKMLHAFDYGNEDISGNVDSFWLDGGIRISATEQISFLRKLYHNKLHVSERSQRIVKQAMLTEANGDYIIRAKTGYSTRIEPKIGWWVGWVELDDNVWFFAMNMDMPTSDGLGLRQAITKEVLKQEKIIP
3AHC , Knot 326 845 0.86 40 324 779 
MGSSHHHHHHSSGLVPRGSHMTNPVIGTPWQKLDRPVSEEAIEGMDKYWRVTNYMSIGQIYLRSNPLMKEPFTRDDVKHRLVGHWGTTPGLNFLLAHINRLIADHQQNTVFIMGPGHGGPAGTSQSYVDGTYTEYYPNITKDEAGLQKFFRQFSYPGGIPSHFAPETPGSIHEGGELGYALSHAYGAVMNNPSLFVPCIIGDGEAETGPLATGWQSNKLVNPRTDGIVLPILHLNGYKIANPTILARISDEELHDFFRGMGYHPYEFVAGFDNEDHMSIHRRFAELFETIFDEICDIKAAAQTDDMTRPFYPMLIFRTPKGWTCPKFIDGKKTEGSWRAHQVPLASARDTEEHFEVLKGWMESYKPEELFNADGSIKDDVTAFMPKGELRIGANPNANGGVIREDLKLPELDQYEVTGVKEYGHGWGQVEAPRALGAYCRDIIKNNPDSFRIFGPDETASNRLNATYEVTDKQWDNGYLSGLVDEHMAVTGQVTEQLSEHQCEGFLEAYLLTGRHGIWSSYESFVHVIDSMLNQHAKWLEATVREIPWRKPISSVNLLVSSHVWRQDHNGFSHQDPGVTSLLINKTFNNDHVTNIYFATDANMLLAISEKCFKSTNKINAIFAGKQPAPTWVTLDEARAELEAGAAEWKWASNAENNDEVQVVLASAGDVPTQELMAASDALNKMGIKFKVVNVVDLLKLQSRENNDEALTDEEFTELFTADKPVLFAYHSYAQDVRGLIYDRPNHDNFHVVGYKEQGSTTTPFDMVRVNDMDRYALQAAALKLIDADKYADKIDELNAFRKKAFQFAVDNGYDIPEFTDWVYPDVKVDETQMLSATAATAGDNE
2VWQ , Knot 153 357 0.84 40 206 338 
MKAIAVKRGEDRPVVIEKPRPEPESGEALVRTLRVGVDGTDHEVIAGGHGGFPEGEDHLVLGHEAVGVVVDPNDTELEEGDIVVPTVRRPPASGTNEYFERDQPDMAPDGMYFERGIVGAHGYMSEFFTSPEKYLVRIPRSQAELGFLIEPISITEKALEHAYASRSAFDWDPSSAFVLGNGSLGLLTLAMLKVDDKGYENLYCLGRRDRPDPTIDIIEELDATYVDSRQTPVEDVPDVYEQMDFIYEATGFPKHAIQSVQALAPNGVGALLGVPSDWAFEVDAGAFHREMVLHNKALVGSVNSHVEHFEAATVTFTKLPKWFLEDLVTGVHPLSEFEAAFDDDDTTIKTAIEFSTV

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 \(c\).

\(|P_{f(8FAJ_1)}(2) \setminus P_{f(3AHC_1)}(2)|=20\), \(|P_{f(3AHC_1)}(2) \setminus P_{f(8FAJ_1)}(2)|=169\). 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 1:1011110111111011111111001000001010100000011111100000011000100100111110010110011110111100000110101000011010000011011000111100011001101010011011001000101010011101110101000101100100001010000001100111001010011010010000101011111111010001111110101100011110011000110000111
Pair \(Z_2\) Length of longest common subsequence
8FAJ_1,3AHC_1 189 4
8FAJ_1,2VWQ_1 151 4
3AHC_1,2VWQ_1 146 4

Newick tree

 
[
	8FAJ_1:89.31,
	[
		2VWQ_1:73,3AHC_1:73
	]:16.31
]

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 \((0.85)(0.8)(\frac{1110 }{\log_{20} 1110}-\frac{265}{\log_{20}265})=225.\)
Status Protein1 Protein2 d d1/2
Query variables 8FAJ_1 3AHC_1 293 190
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]