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Parikh vectors
5MFC_1 9CIS_1 8THK_1 Letter Amino acid
0 16 14 F Phenylalanine
42 31 16 A Alanine
7 37 17 K Lycine
0 28 14 T Threonine
5 3 3 W Tryptophan
0 13 6 Y Tyrosine
26 19 16 E Glutamic acid
39 7 7 Q Glutamine
2 17 4 H Histidine
13 15 6 P Proline
0 5 4 C Cysteine
18 17 13 N Asparagine
6 36 20 D Aspartic acid
20 38 17 G Glycine
22 35 17 I Isoleucine
45 38 18 L Leucine
1 10 5 M Methionine
27 25 15 S Serine
1 15 20 R Arginine
12 19 14 V Valine 

5MFC_1|Chains A, C|YIIIM5AII|synthetic construct (32630)
>9CIS_1|Chains A, B, C, D|6-phosphofructokinase|Candidatus Prometheoarchaeum syntrophicum (2594042)
>8THK_1|Chain A|Guanine nucleotide-binding protein G(i) subunit alpha-2,Guanine nucleotide-binding protein G(s) subunit alpha isoforms short,Guanine nucleotide-binding protein G(q) subunit alpha|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 \(w=f(c)\)
5MFC , Knot 50 286 0.33 32 65 96 
GPGSELPQMVQQLNSPDQQELQSALRKLSQIASGGNEQIQAVIDAGALPALVQLLSSPNEQILQEALWALSNIASGGNEQIQAVIDAGALPALVQLLSSPNEQILQEALWALSNIASGGNEQIQAVIDAGALPALVQLLSSPNEQILQEALWALSNIASGGNEQIQAVIDAGALPALVQLLSSPNEQILQEALWALSNIASGGNEQIQAVIDAGALPALVQLLSSPNEQILQEALWALSNIASGGNEQKQAVKEAGALEKLEQLQSHENEKIQKEAQEALEKLQSH
9CIS , Knot 177 424 0.84 40 219 403 
MGSSHHHHHHSSGLVPRGSHMRVGILTGGGDCPGLNAVIYGALLRASTEKDKEVDVIGIIKGWKVFAIENISPADVDHYTQKLDIGELDDLHTKGGTMLYTSRTNPFKAIAKLKDPIEKEEKTKEIGLELANKFKTLNIDALITIGGDDTCGVAAAMYQYGNAKVCACPKTIDNDLAGTDFTFGFFSGAQLASNTLDNLTTTAHSHQRIFITEIMGRDAGWLTLYSGLSSGADIILLPETPFDFKKDIVEVLMARANSGYKFHMIACSEGAYPTKESLDRDFSVISQKDIDNLPKDAFGNPELPKLNIADKIQKELNKRDDIKKYFNDRHAHYEIRSVVLGHTMRAGTPNVFDRVLGLRYGWHAMSYIIDGNYGKLSALKGTDIVPVDLIEGSKKGLIDPTSDLIQIRDAMTTVKHKSKEKLFK
8THK , Knot 109 246 0.81 40 167 234 
MGSTVSAEDKAAAERSKMIDKNLREDGEKARRTLRLLLLGADNSGKSTIVKQMRILHGGSGGSGGTSGIFETKFQVDKVNFHMFDVGGQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRKEFVDISTASGDGRHICYPHFTCAVDTENARRIFNDCKDIILQMNLREYNLV

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 \(c\).

\(|P_{f(5MFC_1)}(2) \setminus P_{f(9CIS_1)}(2)|=20\), \(|P_{f(9CIS_1)}(2) \setminus P_{f(5MFC_1)}(2)|=174\). 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 1:1110011011001001000010011001001101100010111011111111011001000110011111001101100010111011111111011001000110011111001101100010111011111111011001000110011111001101100010111011111111011001000110011111001101100010111011111111011001000110011111001101100000110011110010010000000100010011001000
Pair \(Z_2\) Length of longest common subsequence
5MFC_1,9CIS_1 194 5
5MFC_1,8THK_1 168 5
9CIS_1,8THK_1 160 4

Newick tree

 
[
	5MFC_1:94.03,
	[
		8THK_1:80,9CIS_1:80
	]:14.03
]

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 \((0.85)(0.8)(\frac{710 }{\log_{20} 710}-\frac{286}{\log_{20}286})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 5MFC_1 9CIS_1 158 97.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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