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Parikh vectors
8SOW_1 4EXB_1 5BWX_1 Letter Amino acid
2 4 13 M Methionine
33 16 17 S Serine
13 22 31 V Valine
10 19 14 D Aspartic acid
4 2 6 C Cysteine
15 7 19 Q Glutamine
10 11 17 K Lycine
8 33 18 A Alanine
0 15 10 H Histidine
5 8 16 F Phenylalanine
8 20 22 R Arginine
7 8 10 N Asparagine
13 12 14 I Isoleucine
6 1 9 W Tryptophan
23 12 17 T Threonine
11 7 9 Y Tyrosine
6 19 23 E Glutamic acid
37 29 32 G Glycine
20 36 46 L Leucine
10 11 26 P Proline 

8SOW_1|Chain A[auth C]|F10 single chain fragment variable|Homo sapiens (9606)
>4EXB_1|Chains A, B, C, D, E, F|Putative uncharacterized protein|Pseudomonas aeruginosa (208964)
>5BWX_1|Chains A, B|Branched-chain-amino-acid aminotransferase, mitochondrial|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 \(w=f(c)\)
8SOW , Knot 98 241 0.74 38 139 210 
EVQLQESGPGLVAPSQSLSITCTVSGFSLTNYDISWIRQPPGKGLEWLGVIWTGGGTNYNSGFMSRLSITKDNSKSQVFLKMNSLQTDDTAIYYCVRQGRSPYWGQGTLVTVSAGILGSGGGGSGGGGSGGGGSDIQVTQSPSSLSVSLGDRVTITCKASKDIYNRLAWYQQKPGNAPRLLISGATSLETGVPSRFSGSGSGKDYTLTITSLQTEDVATYYCQQSWDTPYTFGGGTKLEIK
4EXB , Knot 121 292 0.78 40 170 274 
MHHHHHHSSGVDLGTENLYFQSMIRDTLHDLHRPLGDTGLAVSPLGLGTVKFGRDQGVKYPSGFTIPDDREAADLLALARDLGINLIDTAPAYGRSEERLGPLLRGQREHWVIVSKVGEEFVDGQSVFDFSAAHTRRSVERSLKRLETDRIELVLVHSDGNDLDILENSEVYPTLAALKREGLIGAYGLSGKTVEGGLRALREGDCAMVTYNLNERAERPVIEYAAAHAKGILVKKALASGHACLGAGQDPVRASFELVFDQPGVAAAIVGTINPLHLAHNVAMAAQALKKA
5BWX , Knot 159 369 0.85 40 215 354 
GSHMASSSFKAADLQLEMTQKPHKKPGPGEPLVFGKTFTDHMLMVEWNDKGWGQPRIQPFQNLTLHPASSSLHYSLQLFEGMKAFKGKDQQVRLFRPWLNMDRMLRSAMRLCLPSFDKLELLECIRRLIEVDKDWVPDAAGTSLYVRPVLIGNEPSLGVSQPTRALLFVILCPVGAYFPGGSVTPVSLLADPAFIRAWVGGVGNYKLGGNYGPTVLVQQEALKRGCEQVLWLYGPDHQLTEVGTMNIFVYWTHEDGVLELVTPPLNGVILPGVVRQSLLDMAQTWGEFRVVERTITMKQLLRALEEGRVREVFGSGTACQVCPVHRILYKDRNLHIPTMENGPELILRFQKELKEIQYGIRAHEWMFPV

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 \(c\).

\(|P_{f(8SOW_1)}(2) \setminus P_{f(4EXB_1)}(2)|=71\), \(|P_{f(4EXB_1)}(2) \setminus P_{f(8SOW_1)}(2)|=102\). 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 1:0101000111111100010100010110100001011001110110111111011100000111001010000000011101001000001100010010010110101101011111011110111101111001010001001010110010100010001000111000011011011101100100111001010101000010100100001100000001001001111001010
Pair \(Z_2\) Length of longest common subsequence
8SOW_1,4EXB_1 173 3
8SOW_1,5BWX_1 184 3
4EXB_1,5BWX_1 159 4

Newick tree

 
[
	8SOW_1:92.32,
	[
		4EXB_1:79.5,5BWX_1:79.5
	]:12.82
]

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 \((0.85)(0.8)(\frac{533 }{\log_{20} 533}-\frac{241}{\log_{20}241})=83.4\)
Status Protein1 Protein2 d d1/2
Query variables 8SOW_1 4EXB_1 104 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]