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Parikh vectors
7DMW_1 3LFI_1 6PAE_1 Letter Amino acid
31 52 28 L Leucine
8 17 13 Y Tyrosine
13 27 19 R Arginine
2 1 0 C Cysteine
24 16 20 I Isoleucine
10 34 6 F Phenylalanine
12 33 12 P Proline
18 49 25 T Threonine
7 18 0 W Tryptophan
6 28 14 N Asparagine
14 58 32 G Glycine
23 10 16 K Lycine
17 34 21 D Aspartic acid
20 29 8 Q Glutamine
10 12 6 H Histidine
14 51 18 S Serine
17 59 35 V Valine
15 59 31 A Alanine
24 45 14 E Glutamic acid
8 2 9 M Methionine 

7DMW_1|Chains A, B, C, D, E|CcpC|Bacillus velezensis FZB42 (326423)
>3LFI_1|Chains A, B|Fructosyltransferase|Aspergillus japonicus (34381)
>6PAE_1|Chains A, B, C, D|L-asparaginase|Dickeya chrysanthemi (556)
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 \(w=f(c)\)
7DMW , Knot 129 293 0.83 40 191 281 
MQLQELHMLVVLAEELNMRKAAERLFVSQPALSQRLQTIEKAWGTKIFLRSQKGLTVTPAGEKIIQFANDVTDRQERIRENIDELEGEIHGTLKLAVASIIGQHWLPKVLKTYVERYPNAKVSLITGWSSEMLKSLYEDQVHIGIIRGNPEWKGRKDYLMTDHLYLVDTEISCIDDIAHTDRPFIQFKSDSTYFQEIQHWWHQKFKTSPKQTILVDQIETCKQMALHGIGYAILPSVTLEEEDKVNKMPLLDTKDHPIGRDTWLLGYEPAFELKQVQAFVSVIKDMLKQERPF
3LFI , Knot 249 634 0.84 40 253 585 
SYHLDTTAPPPTNLSTLPNNTLFHLWRPRAHILPAEGQIGDPCAHYTDPSTGLFHVGFLHDGDGIAGATTANLATYTDTSDNGSFLIQPGGKNDPVAVFDGAVIPVGVNNTPTLLYTSVSFLPIHWSIPYTRGSETQSLAVARDGGRRFDKLDQGPVIADHPFAVDVTAFRDPFVFRSARLDVLLSLDEEVARNETAVQQAVDGWTEKNAPWYVAVSGGVHGVGPAQFLYRQNGGNASEFQYWEYLGEWWQEATNSSWGDEGTWAGRWGFNFETGNVLFLTEEGHDPQTGEVFVTLGTEGSGLPIVPQVSSIHDMLWAAGEVGVGSEQEGAKVEFSPSMAGFLDWGFSAYAAAGKVLPASSAVSKTSGVEVDRYVSFVWLTGDQYEQADGFPTAQQGWTGSLLLPRELKVQTVENVVDNELVREEGVSWVVGESDNQTATLRTLGITIARETKAALLANGSVTAEEDRTLQTAAVVPFAQSPSSKFFVLTAQLEFPASARSSPLQSGFEILASELERTAIYYQFSNESLVVDRSQTSAAAPTNPGLDSFTESGKLRLFDVIENGQEQVETLDLTVVVDNAVVEVYANGRFALSTWARSWYDNSTQIRFFHNGEGEVQFRNVSVSEGLYNAWPER
6PAE , Knot 138 327 0.81 36 184 310 
ADKLPNIVILATGGTIAGSAATGTQTTGYKAGALGVDTLINAVPEVKKLANVKGEQFSNMASENMTGDVVLKLSQRVNELLARDDVDGVVITHGTDTVEESAYFLHLTVKSDKPVVFVAAMRPATAISADGPMNLLEAVRVAGDKQSRGRGVMVVINDRIGSARYITKTNASTLDTFRANEEGYLGVIIGNRIYYQNRIDKLHTTRSVFDVRGLTSLPKVDILYGYQDDPEYLYDAAIQHGVKGIVYAGMGAGSVSVRGIAGMRKALEKGVVVMRSTRTGNGIVPPDEELPGLVSDSLNPAHARILLMLALTRTSDPKVIQEYFHTY

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 \(c\).

\(|P_{f(7DMW_1)}(2) \setminus P_{f(3LFI_1)}(2)|=46\), \(|P_{f(3LFI_1)}(2) \setminus P_{f(7DMW_1)}(2)|=108\). 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 1:10100101111110010100110011100111000100100111001110000110101110011011001000000100010010101010101111011100111011000100010101011011000110010000101111010101010000110001011000100100110000111010000001001001100010001000111001000001110111011110101000001001111000001110001111001110100101110110011000011
Pair \(Z_2\) Length of longest common subsequence
7DMW_1,3LFI_1 154 4
7DMW_1,6PAE_1 149 4
3LFI_1,6PAE_1 159 4

Newick tree

 
[
	3LFI_1:79.47,
	[
		7DMW_1:74.5,6PAE_1:74.5
	]:4.97
]

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 \((0.85)(0.8)(\frac{927 }{\log_{20} 927}-\frac{293}{\log_{20}293})=171.\)
Status Protein1 Protein2 d d1/2
Query variables 7DMW_1 3LFI_1 216 158
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]