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Parikh vectors
9EMN_1 1VHK_1 8PUM_1 Letter Amino acid
38 18 42 A Alanine
23 17 17 Q Glutamine
38 18 34 G Glycine
25 9 8 F Phenylalanine
32 9 20 P Proline
33 21 22 S Serine
23 12 14 T Threonine
27 11 17 D Aspartic acid
4 6 8 C Cysteine
14 10 21 H Histidine
21 18 5 I Isoleucine
51 22 45 L Leucine
21 19 7 K Lycine
11 3 4 W Tryptophan
23 5 8 N Asparagine
18 5 10 M Methionine
14 6 7 Y Tyrosine
42 13 28 R Arginine
37 28 26 E Glutamic acid
37 18 39 V Valine 

9EMN_1|Chains A, B, C, D|Glucose-6-phosphate 1-dehydrogenase|Synechocystis sp. (strain PCC 6803 / Kazusa) (1111708)
>1VHK_1|Chains A, B, C, D|Hypothetical protein yqeU|Bacillus subtilis (1423)
>8PUM_1|Chains A, B|L-threonine aldolase|Mus musculus (10090)
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 \(w=f(c)\)
9EMN , Knot 212 532 0.83 40 260 505 
MGSSHHHHHHSSGLVPRGSHMLEMVTLLENPFRTGLRQERTPEPLILTIFGASGDLTQRKLVPAIYQMKRERRLPPELTVVGFARRDWSHDHFREQMRKGIEEFSTGIGSEDLWNEFAQGLFYCSGNMDDPESYLKLKNFLGELDEKRNTRGNRVFYLAVSPNFFPPGIKQLGAAGMLSDPVKSRIVIEKPFGRDLSSAQSLNRVVQSVCKENQVYRIDHYLGKETVQNLMVFRFANAIFEPLWNRQFVDHVQITVAETVGVEERAGYYESAGALRDMVQNHLMQLFCLTAMDPPNAIDADSIRNEKVKVLQATRLADINNLENAGIRGQYKAGWMGGKPVPGYREEPGVDPSSTTPTFAALKLMVDNWRWQGVPFYLRTGKRMPKKVSEIAIQFRQVPLLIFQSVAHQANPNVLSLRIQPNEGISLRFEAKMPGSELRTRTVDMDFSYGSSFGVAAADAYHRLLLDCMLGDQTLFTRADEVEEAWRVVTPVLSAWDAPSDPLSMPLYEAGTWEPAEAEWLINKDGRRWRRL
1VHK , Knot 117 268 0.81 40 166 253 
MSLQRYFIELTKQQIEEAPTFSITGEEVHHIVNVMRMNEGDQIICCSQDGFEAKCELQSVSKDKVSCLVIEWTNENRELPIKVYIASGLPKGDKLEWIIQKGTELGAHAFIPFQAARSVVKLDDKKAKKKRERWTKIAKEAAEQSYRNEVPRVMDVHSFQQLLQRMQDFDKCVVAYEESSKQGEISAFSAIVSSLPKGSSLLIVFGPEGGLTEAEVERLTEQDGVTCGLGPRILRTETAPLYALSAISYQTELLRGDQEGGSHHHHHH
8PUM , Knot 159 382 0.82 40 195 357 
MGSSHHHHHHSSGENLYFQGHMVDLRSDTVTRPGPAMRRAMAEAVVGDDDYGEDPTVHELQEKAAELLGVERTLFVPTNTMANLISVMGHCRRRGSQVLLGQECHLHVYEQGGVAQIAGVHSHPLPDLPYGTLDLNELERALTRGSGSPYHPVCELVYLENTHSSAGGRVLPVDYLRQVCLLAHAHGARVHMDGARLMNAAVALRIPPARLVEHCDSVSFCFSKGLGAPVGALVGGSKDFIGEAWRLRKALGGGMRQAGVLAAAALVGLAEAEEVLPRDHENAQRFAKGLQDLASPICSVDPATVETNMVLVQVAGLPPSELCQRLQAVSAEEVAQTGRAVRVLLFPWTEQSVRAVWHRDVSAQDTELALKKWEFVLRQLRP

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 \(c\).

\(|P_{f(9EMN_1)}(2) \setminus P_{f(1VHK_1)}(2)|=134\), \(|P_{f(1VHK_1)}(2) \setminus P_{f(9EMN_1)}(2)|=40\). 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 1:1100000000001111010011011011001100110000010111101111010100001111100100000111010111110001000010001001100100111000110011011100010100100010100111010000000100110111010111111001111111001100011100111001001001001100100000100100011000100111101101110111000110010101100111000110000111100110001101101011011011010010000101101001101001001110100011111101111000011101000010111101110010101111010010011001001110100111111001100101011010101001101010101110010000101010010011111101000111001110001100100100110110111011011001101110011010110101110001001001
Pair \(Z_2\) Length of longest common subsequence
9EMN_1,1VHK_1 174 7
9EMN_1,8PUM_1 157 13
1VHK_1,8PUM_1 155 7

Newick tree

 
[
	9EMN_1:84.56,
	[
		8PUM_1:77.5,1VHK_1:77.5
	]:7.06
]

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 \((0.85)(0.8)(\frac{800 }{\log_{20} 800}-\frac{268}{\log_{20}268})=146.\)
Status Protein1 Protein2 d d1/2
Query variables 9EMN_1 1VHK_1 182 135.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]