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Parikh vectors
4PTH_1 2HLF_1 4CZQ_1 Letter Amino acid
11 64 37 L Leucine
5 6 1 W Tryptophan
13 52 36 A Alanine
12 16 15 E Glutamic acid
5 6 8 H Histidine
5 17 5 M Methionine
6 27 24 F Phenylalanine
10 19 30 P Proline
6 25 26 T Threonine
4 8 0 Y Tyrosine
9 21 10 R Arginine
2 3 10 C Cysteine
12 37 14 I Isoleucine
11 30 26 V Valine
6 11 11 N Asparagine
6 11 7 K Lycine
9 11 30 S Serine
13 10 33 D Aspartic acid
4 13 15 Q Glutamine
10 57 31 G Glycine 

4PTH_1|Chain A|Dihydrofolate reductase|Escherichia coli (316385)
>2HLF_1|Chains A, B|H(+)/Cl(-) exchange transporter clcA|Escherichia coli (562)
>4CZQ_1|Chain A|EXTRALONG MANGANESE PEROXIDASE|CERIPORIOPSIS SUBVERMISPORA (42742)
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 \(w=f(c)\)
4PTH , Knot 81 159 0.86 40 124 153 
MISLIAALAVDRVIGMENAMPWNLPADLAWFKRNTLNKPVIMGRHTWESIGRPLPGRKNIILSSQPGTDDRVTWVKSVDEAIAACGDVPEIMVIGGGRVYEQFLPKAQKLYLTHIDAEVEGDTHFPDYEPDDWESVFSEFHDADAQNSHSYCFEILERR
2HLF , Knot 173 444 0.79 40 205 398 
RRRQLIRQLLERDKTPLAILFMAAVVGTLVGLAAVAFDKGVAWLQNQRMGALVHTADNYPLLLTVAFLCSAVLAMFGYFLVRKYAPEAGGSGIPEIEGALEDQRPVRWWRVLPVKFFGGLGTLGGGMVLGREGPTVQIGGNIGRMVLDIFRLKGDEARHTLLATGAAAGLAAAFNAPLAGILFIIEEMRPQFRYTLISIKAVFIGVIMSTIMYRIFNHEVALIDVGKLSDAPLNTLWLYLILGIIFGIFGPIFNKWVLGMQDLLHRVHGGNITKWVLMGGAIGGLCGLLGFVAPATSGGGFNLIPIATAGNFSMGMLVFIFVARVITTLLCFSSGAPGGIFAPMLALGTVLGTAFGMVAVELFPQYHLEAGTFAIAGMGALLAASIRAPLTGIILVLEMTDNYQLILPMIITGLGATLLAQFTGGKPLESAILARTLAKQEAEQ
4CZQ , Knot 151 369 0.80 38 185 343 
MAPTAVCSDGTRVSNAVCCDFVSLGQDLQSMVLQGDCGEDAHEIIRLTFHDAVAISRKLGPSAGGGADGSMLLFPLVEPEFAASNGIDDSVNNLIPFLSLHPTISAGDLVQFAGAVALSNCPGAPRVQFLAGRPNHTIAAIDGLIPEPQDNVTSILERFDDAGGFTPFEVVSLLASHTIARADKVDPTLDAAPFDTTPFTFDSQIFLEVLLKGVGFPGLDNNTGEVSSPLPLGDTSTGGKDTGLMRLQSDFALAHDPRTACFWQGFVDQQEFMSQSFASAFAKLAVLGHNTDDLIDCSEVVPVPKPAVDKPTTFPATTGPQDLELSCLAERFPTLSVDPGAQETLIPHCSDGLENCTSVQFSGPATDSP

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 \(c\).

\(|P_{f(4PTH_1)}(2) \setminus P_{f(2HLF_1)}(2)|=45\), \(|P_{f(2HLF_1)}(2) \setminus P_{f(4PTH_1)}(2)|=126\). 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 1:110111111100111100111101110111100001001111100010011011110001110001100001011001001111010110111111101000111010010100101010100011000100100110010010100000001011000
Pair \(Z_2\) Length of longest common subsequence
4PTH_1,2HLF_1 171 3
4PTH_1,4CZQ_1 167 4
2HLF_1,4CZQ_1 144 4

Newick tree

 
[
	4PTH_1:88.28,
	[
		4CZQ_1:72,2HLF_1:72
	]:16.28
]

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 \((0.85)(0.8)(\frac{603 }{\log_{20} 603}-\frac{159}{\log_{20}159})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 4PTH_1 2HLF_1 152 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]