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Parikh vectors
4DXP_1 3AHI_1 7NRF_1 Letter Amino acid
8 34 11 R Arginine
5 28 17 Q Glutamine
21 41 16 K Lycine
13 31 32 Y Tyrosine
15 34 11 H Histidine
14 62 21 L Leucine
12 44 18 P Proline
2 17 10 W Tryptophan
9 70 22 A Alanine
21 64 44 G Glycine
11 43 11 I Isoleucine
13 40 24 F Phenylalanine
8 21 10 M Methionine
8 45 39 S Serine
13 50 30 T Threonine
14 51 22 V Valine
9 46 26 N Asparagine
14 59 36 D Aspartic acid
4 6 0 C Cysteine
16 59 11 E Glutamic acid 

4DXP_1|Chain A|LEA X121 GFP-LIKE PROTEINS|Synthetic Construct (32630)
>3AHI_1|Chain A|Xylulose 5-phosphate/fructose 6-phosphate phosphoketolase|Bifidobacterium breve (1685)
>7NRF_1|Chain A|Outer membrane protein assembly factor BamA|Escherichia coli O157:H7 (83334)
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 \(w=f(c)\)
4DXP , Knot 105 230 0.82 40 170 219 
MSVIKSDMKIKLRMEGTVNGHKFVIEGEGEGKPYEGTQTMNLKVKEGAPLPFAYDILTTVFHYGNRVFAKYPKHIPDYFKQSFPEGYSWERSMTFEDGGICTARNDITLEGDCFFYEIRFDGVNFPPNGPVMQKKTLKWEPSTEKMYVRDGVLTGDINMALLLEGGGHYRCDFKTTYKAKKGVQLPDYHFVDHCIEILSHDKDYNNVKLYEHAVAHSGLPRQAKHHHHHH
3AHI , Knot 326 845 0.86 40 324 778 
MGSSHHHHHHSSGLVPRGSHMTNPVIGTPWQKLDRPVSEEAIEGMDKYWRVTNYMSIGQIYLRSNPLMKEPFTRDDVKHRLVGHWGTTPGLNFLLAHINRLIADHQQNTVFIMGPGHGGPAGTSQSYVDGTYTEYYPNITKDEAGLQKFFRQFSYPGGIPSHFAPETPGSIHEGGELGYALSHAYGAVMNNPSLFVPCIIGDGEAETGPLATGWQSNKLVNPRTDGIVLPILHLNGYKIANPTILARISDEELHDFFRGMGYHPYEFVAGFDNEDHMSIHRRFAELFETIFDEICDIKAAAQTDDMTRPFYPMLIFRTPKGWTCPKFIDGKKTEGSWRAAQVPLASARDTEEHFEVLKGWMESYKPEELFNADGSIKDDVTAFMPKGELRIGANPNANGGVIREDLKLPELDQYEVTGVKEYGHGWGQVEAPRALGAYCRDIIKNNPDSFRIFGPDETASNRLNATYEVTDKQWDNGYLSGLVDEHMAVTGQVTEQLSEHQCEGFLEAYLLTGRHGIWSSYESFVHVIDSMLNQHAKWLEATVREIPWRKPISSVNLLVSSHVWRQDHNGFSHQDPGVTSLLINKTFNNDHVTNIYFATDANMLLAISEKCFKSTNKINAIFAGKQPAPTWVTLDEARAELEAGAAEWKWASNAENNDEVQVVLASAGDVPTQELMAASDALNKMGIKFKVVNVVDLLKLQSRENNDEALTDEEFTELFTADKPVLFAYHSYAQDVRGLIYDRPNHDNFHVVGYKEQGSTTTPFDMVRVNDMDRYALQAAALKLIDADKYADKIDELNAFRKKAFQFAVDNGYDIPEFTDWVYPDVKVDETQMLSATAATAGDNE
7NRF , Knot 171 411 0.83 38 218 389 
MGSSHHHHHHSSGENLYFQHMRNTGSFNFGIGYGTESGVSFQAGVQQDNWLGTGYAVGINGTKNDYQTYAELSVTNPYFTVDGVSLGGRLFYNDFQADDADLSDYTNKSYGTDVTLGFPINEYNSLRAGLGYVHNSLSNMQPQVAMWRYLYSMGEHPSTSDQDNSFKTDDFTFNYGWTYNKLDRGYFPTDGSRVNLTGKVTIPGSDNEYYKVTLDTATYVPIDDDHKWVVLGRTRWGYGDGLGGKEMPFYENFYAGGSSTVRGFQSNTIGPKAVYFPHQASNYDPDYDYESATQDGAKDLSKSDDAVGGNAMAVASLEFITPTPFISDKYANSVRTSFFWDMGTVWDTNWDSSQYSGYPDYSDPSNIRMSAGIALQWMSPLGPLVFSYAQPFKKYDGDKAEQFQFNIGKTW

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 \(c\).

\(|P_{f(4DXP_1)}(2) \setminus P_{f(3AHI_1)}(2)|=21\), \(|P_{f(3AHI_1)}(2) \setminus P_{f(4DXP_1)}(2)|=175\). 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 1:10110001010101010101001110101010100100010101001111111001100110010011100100110010001101001000101001110010001010100110010101101110111100001010100001010011101010111110111000001000001001101100011000101100000000101000111001110010000000
Pair \(Z_2\) Length of longest common subsequence
4DXP_1,3AHI_1 196 6
4DXP_1,7NRF_1 190 6
3AHI_1,7NRF_1 152 13

Newick tree

 
[
	4DXP_1:10.44,
	[
		7NRF_1:76,3AHI_1:76
	]:26.44
]

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 \((0.85)(0.8)(\frac{1075 }{\log_{20} 1075}-\frac{230}{\log_{20}230})=227.\)
Status Protein1 Protein2 d d1/2
Query variables 4DXP_1 3AHI_1 292 183.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]