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Parikh vectors
5SAC_1 3QLD_1 2FZU_1 Letter Amino acid
9 35 6 R Arginine
6 12 10 H Histidine
6 9 5 M Methionine
21 17 17 T Threonine
17 48 10 A Alanine
29 43 20 L Leucine
20 16 11 S Serine
11 8 11 Y Tyrosine
17 3 13 N Asparagine
23 22 16 E Glutamic acid
23 29 22 G Glycine
23 12 12 I Isoleucine
25 9 20 K Lycine
14 19 10 P Proline
23 29 18 D Aspartic acid
5 6 2 C Cysteine
15 15 8 Q Glutamine
21 9 11 F Phenylalanine
3 9 1 W Tryptophan
38 38 16 V Valine 

5SAC_1|Chains A, B|Uridylate-specific endoribonuclease|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3QLD_1|Chains A, B|Mandelate racemase/muconate lactonizing protein|Alicyclobacillus acidocaldarius LAA1 (543302)
>2FZU_1|Chain A|Green fluorescent protein|Aequorea victoria (6100)
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 \(w=f(c)\)
5SAC , Knot 146 349 0.81 40 200 330 
GAMSLENVAFNVVNKGHFDGQQGEVPVSIINNTVYTKVDGVDVELFENKTTLPVNVAFELWAKRNIKPVPEVKILNNLGVDIAANTVIWDYKRDAPAHISTIGVCSMTDIAKKPTETICAPLTVFFDGRVDGQVDLFRNARNGVLITEGSVKGLQPSVGPKQASLNGVTLIGEAVKTQFNYYKKVDGVVQQLPETYFTQSRNLQEFKPRSQMEIDFLELAMDEFIERYKLEGYAFEHIVYGDFSHSQLGGLHLLIGLAKRFKESPFELEDFIPMDSTVKNYFITDAQTGSSKCVCSVIDLLLDDFVEIIKSQDLSVVSKVVKVTIDYTEISFMLWCKDGHVETFYPKLQ
3QLD , Knot 160 388 0.82 40 194 359 
MSLQTCVLHRLSLPLKFPMRTAHGHIREKQAILVQLVDADGIEGWSECVALAEPTYTEECTDTAWVMLVHHLVPRFARWLRAASQDQDVDPRTVCEALRDVRGNRMSVAAIEMAVWDWYAARTGQPLVGLLGGGRDRVEVSATLGMSESLDVLIQSVDAAVEQGFRRVKLKIAPGRDRAAIKAVRLRYPDLAIAADANGSYRPEDAPVLRQLDAYDLQFIEQPLPEDDWFDLAKLQASLRTPVCLDESVRSVRELKLTARLGAARVLNVKPGRLGGFGATLRALDVAGEAGMAAWVGGMYETGVGRVHGLIAAALPLMRYATDLGPSDRYFEQDVLKEPIAFVEPGVIQVPQCAGVADWVDRDAVRRFSTATWSVDLRTLEGHHHHHH
2FZU , Knot 110 239 0.84 40 161 227 
MASKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKQHDFFKSAMPEGYVQERTISFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNYNSHNVYITADKQKNGIKANFKIRHNIEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSALSKDPNEKRDHMVLLEFVTAAGITHGMDELYK

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 \(c\).

\(|P_{f(5SAC_1)}(2) \setminus P_{f(3QLD_1)}(2)|=86\), \(|P_{f(3QLD_1)}(2) \setminus P_{f(5SAC_1)}(2)|=80\). 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 1:1110100111011001010100101110110001000101101011000001110111011100010111010110011101110011100000111010011100100110010001011101110101010101100100111100101011010111001010110111011000100000101110011000100000100101000101011011100110000101011001101010000111101111110010001101001111000100011001001000010011011100110110000101100110101000010111100010100101010
Pair \(Z_2\) Length of longest common subsequence
5SAC_1,3QLD_1 166 4
5SAC_1,2FZU_1 167 3
3QLD_1,2FZU_1 183 4

Newick tree

 
[
	2FZU_1:89.06,
	[
		5SAC_1:83,3QLD_1:83
	]:6.06
]

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 \((0.85)(0.8)(\frac{737 }{\log_{20} 737}-\frac{349}{\log_{20}349})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 5SAC_1 3QLD_1 137 129.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]