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Parikh vectors
1QWI_1 6DRJ_1 7JHU_1 Letter Amino acid
5 59 0 P Proline
12 66 2 T Threonine
8 91 0 V Valine
2 89 0 R Arginine
2 34 2 C Cysteine
10 88 0 I Isoleucine
14 85 4 G Glycine
3 33 0 M Methionine
11 107 0 S Serine
3 51 0 N Asparagine
8 78 0 D Aspartic acid
9 107 0 E Glutamic acid
16 88 4 A Alanine
3 44 0 H Histidine
10 167 0 L Leucine
1 35 0 W Tryptophan
2 40 0 Y Tyrosine
6 64 0 Q Glutamine
12 84 0 K Lycine
6 63 0 F Phenylalanine 

1QWI_1|Chains A, B, C, D|osmotically inducible protein|Escherichia coli (562)
>6DRJ_1|Chains A, B, C, D|Transient receptor potential cation channel, subfamily M, member 2|Danio rerio (7955)
>7JHU_1|Chain A|DNA (5'-D(*GP*AP*GP*CP*AP*GP*AP*CP*TP*TP*GP*A)-3')|synthetic construct (32630)
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 \(w=f(c)\)
1QWI , Knot 74 143 0.85 40 115 138 
MTIHKKGQAHWESDIKRGKGTVSTESGVLNQQPYGFNTRFEGEKGTNPEELIGAAHAACFSMALSLMLGEAGFTPTSIDTTADVSLDKVDAGFAITKIALKSEVAVPGIDASTFDGIIQKAKAGCPVSQVLKAEITLDYQLKS
6DRJ , Knot 519 1473 0.85 40 362 1277 
MLGTSGVKIHPNGNSNQLGVQLENVKLTSLFKKLDKRCSLASWIKENIKKKECCFYVEDGREGICKCGYPKVQHCDEAIKPEDYMGEQWDKHRHVRETPTDAFGDISFGGLGQKTGKYVRVSSDTSCENLYQLMTEQWKLRSPNLLISVTGGAKNFYIKTHLKDKFRRGLIKVAQTTGAWILTGGTHAGVMKHVGMAVRDYTLSSGSMEGQIVVIGVAPWGVIHNRSTLIHPEGRFPAYYSLDEQGQGRLSCLDINHTHFLLVDDGTQGHYGVEIELRARLEKLISKLSLGNRESGVTIPVVCVVLDGGPGTLNTIYNSMLNHTPCVVLEGSGRLADVIAHVASVPVSKVTMALINRLLKRFFMQEYKNFTELQIIEWTKKIQDILRMPHLLTVFRIDEDKNYDVDVAILQALLKASRSDEHAGRHCWERQLELAVAWNRVDIAESEIFTEESQWTSSDLHPAMFSALVGDKPEFVRLLLENGVCVREFLEREETLCELYSHLPSCFFLRKLAKRVQGGKMRRGQEPLPGSRKVCLSHVSEEVRHLLGSFTQPLYIASRYKPTKDDVRLKVPSKGALDLPCSGEEWSADTVWDPGRDLFLWAVVQNNRELAEIGWEQCRDCIAAALAASKILRKLAQESGEDDSEEATEMLELANHYEKQAIGVFSECHSWDAQRAQKLLIRISPSWGRSTCLWLALEAHDKSFIAHSGVQALLTQIWCGELSVDNPHWKVLLCMIFFPLIYTGFLTFRRDEDIQRQAERTQQKLAMESVFAGQSDGKIKRHLRGFSQKSELKPLNCSSRLMSFLKSPQVKFYWNIASYFGFLWLFAVVLMIDFQTSPSWRELLLYVWLTSLVCEEIRQLYHDFDGSGFRRKAKMYIKDLWNILDVLSIVLFIAGLICRLQASDTVFYIGKVILCIDFIIFCLRLMAIFSISRTLGPKIIIVRRMMLDLFFFMFLLSIWVVAYGVAKQGILIENEERLNWIIRGAVYEPYITIFGNFPTNIDNTLFDISSCSVNASDPLKPKCPMLNADNTPVFPEWLTIMMLCVYLLFANILLLNLLIAIFNYTFQEVQDNTDTIWKFQRYELIKEYHSRPALPPPFILLSHLILFIRGVFLRDLPQRHKNFRQELEQTEEEELLSWEAYMKDNYLASTRQDESQSVEHRIHDTAEKVGAMSELLEREQEMVSATMAKRLARLEEQVSESAKALRWIIDALKSQGCKSKVQPPLMRSKSSDRDDGDSSGQETDDEEAPHMFARQLQYPDSTVRRFPVPEEKVSWEVNFSPYQPPVYNQQDSSESDTSALDKHRNPGGRTGIRGKGALNTLGPNHILHPIFTRWRDAEHKVLEFLAVWEDAEKRWALLGGPAQPDEPLAQVLERILGKKLNEKTKTLLKAGEEVYKGYVDDSRNTDNAWVETSIITLHCDKNTPLMADLNHMVESSLSSHQPLQWREVSSDACRCSYQREALRQIAHHHNTYF
7JHU , Knot 7 12 0.48 8 8 10 
GAGCAGACTTGA

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 \(c\).

\(|P_{f(1QWI_1)}(2) \setminus P_{f(6DRJ_1)}(2)|=1\), \(|P_{f(6DRJ_1)}(2) \setminus P_{f(1QWI_1)}(2)|=248\). 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 1:10100010101000100101010000111000101100010100100100111110110101110111101110100100010101001011111001110001111110100101110010110110011010101000100
Pair \(Z_2\) Length of longest common subsequence
1QWI_1,6DRJ_1 249 4
1QWI_1,7JHU_1 113 3
6DRJ_1,7JHU_1 358 4

Newick tree

 
[
	6DRJ_1:17.01,
	[
		1QWI_1:56.5,7JHU_1:56.5
	]:11.51
]

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 \((0.85)(0.8)(\frac{1616 }{\log_{20} 1616}-\frac{143}{\log_{20}143})=386.\)
Status Protein1 Protein2 d d1/2
Query variables 1QWI_1 6DRJ_1 495 269.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]