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Parikh vectors
5SMA_1 5KUJ_1 8FZB_1 Letter Amino acid
34 6 33 A Alanine
16 6 10 Q Glutamine
20 3 11 H Histidine
30 4 11 K Lycine
10 0 5 W Tryptophan
22 10 22 R Arginine
25 6 10 I Isoleucine
30 3 10 F Phenylalanine
29 10 22 S Serine
43 14 24 V Valine
27 4 7 N Asparagine
23 1 8 C Cysteine
15 4 29 E Glutamic acid
26 6 16 P Proline
25 4 12 Y Tyrosine
34 12 12 D Aspartic acid
29 6 16 G Glycine
40 16 46 L Leucine
14 1 5 M Methionine
31 8 22 T Threonine 

5SMA_1|Chain A[auth D]|Proofreading exoribonuclease nsp14|Severe acute respiratory syndrome coronavirus 2 (2697049)
>5KUJ_1|Chain A|Calcium uniporter protein, mitochondrial|Homo sapiens (9606)
>8FZB_1|Chains A, B, C|Protein-lysine N-methyltransferase EEF2KMT|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 \(w=f(c)\)
5SMA , Knot 217 523 0.86 40 267 499 
SMLFKDCSKVITGLHPTQAPTHLSVDTKFKTEGLCVDIPGIPKDMTYRRLISMMGFKMNYQVNGYPNMFITREEAIRHVRAWIGFDVEGCHATREAVGTNLPLQLGFSTGVNLVAVPTGYVDTPNNTDFSRVSAKPPPGDQFKHLIPLMYKGLPWNVVRIKIVQMLSDTLKNLSDRVVFVLWAHGFELTSMKYFVKIGPERTCCLCDRRATCFSTASDTYACWHHSIGFDYVYNPFMIDVQQWGFTGNLQSNHDLYCQVHGNAHVASCDAIMTRCLAVHECFVKRVDWTIEYPIIGDELKINAACRKVQHMVVKAALLADKFPVLHDIGNPKAIKCVPQADVEWKFYDAQPCSDKAYKIEELFYSYATHSDKFTDGVCLFWNCNVDRYPANSIVCRFDTRVLSNLNLPGCDGGSLYVNKHAFHTPAFDKSAFVNLKQLPFFYYSDSPCESHGKQVVSDIDYVPLKSATCITRCNLGGAVCRHHANEYRLYLDAYNMMISAGFSLWVYKQFDTYNLWNTFTRLQ
5KUJ , Knot 62 124 0.80 38 95 118 
GSHMASSDDVTVVYQNGLPVISVRLPSRRERCQFTLKPISDSVGVFLRQLQEEDRGIDRVAIYSPDGVRVAASTGIDLLLLDDFKLVINDLTYHVRPPKRDLLSHENAATLNDVKTLVQQLYTT
8FZB , Knot 138 331 0.80 40 191 314 
GMAPEENAGTELLLQSFERRFLAARTLRSFPWQSLEAKLRDSSDSELLRDILHKTVKHPVCVKHPPSVKYARCFLSELIKKHEAVHTEPLDELYEALAETLMAKESTQGHRSYLLPSGGSVTLSESTAIISYGTTGLVTWDAALYLAEWAIENPAVFTNRTVLELGSGAGLTGLAICKMCRPRAYIFSDCHSRVLEQLRGNVLLNGLSLEADITAKLDSPRVTVAQLDWDVATVHQLSAFQPDVVIAADVLYCPEAYYSLVGVLRRLAACREHQRAPEVYVAFTVRNPETCQLFTTYLGRAGIRWEVEPRHEQKLFPYEEHLEMAMLNLTL

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 \(c\).

\(|P_{f(5SMA_1)}(2) \setminus P_{f(5KUJ_1)}(2)|=195\), \(|P_{f(5KUJ_1)}(2) \setminus P_{f(5SMA_1)}(2)|=23\). 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 1:0111000001101101001100101000100011010111110010000110111101000101010111000011001011111010100100011100111011100110111110101001000010010101111001001111100111101101011011000100100011111110110100100110111000001000010010010000101000111001001111010011101010000010001010101100011100011100011001010100111100101011000100111011111001111001101011001101010101001010000100100110001000001001101110001000110011001000110010111001101010001100111000111010011110000010000100110010011100100100001111100001000010101001110111011100010000110010010
Pair \(Z_2\) Length of longest common subsequence
5SMA_1,5KUJ_1 218 3
5SMA_1,8FZB_1 194 4
5KUJ_1,8FZB_1 154 3

Newick tree

 
[
	5SMA_1:11.53,
	[
		8FZB_1:77,5KUJ_1:77
	]:33.53
]

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 \((0.85)(0.8)(\frac{647 }{\log_{20} 647}-\frac{124}{\log_{20}124})=151.\)
Status Protein1 Protein2 d d1/2
Query variables 5SMA_1 5KUJ_1 195 118.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]