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Parikh vectors
1ZBB_1 5LGE_1 1TCV_1 Letter Amino acid
0 7 1 W Tryptophan
0 25 31 V Valine
98 31 17 A Alanine
0 18 9 Q Glutamine
0 30 16 E Glutamic acid
0 30 21 I Isoleucine
97 24 13 T Threonine
0 23 15 S Serine
76 5 5 C Cysteine
76 33 28 G Glycine
0 28 29 L Leucine
0 37 17 K Lycine
0 17 9 F Phenylalanine
0 17 17 N Asparagine
0 14 10 M Methionine
0 17 7 Y Tyrosine
0 14 9 R Arginine
0 25 11 D Aspartic acid
0 18 9 H Histidine
0 12 13 P Proline 

1ZBB_1|Chain A[auth I]|DNA STRAND 1 (ARBITRARY MODEL SEQUENCE)|null
>5LGE_1|Chains A, B, C, D|Isocitrate dehydrogenase [NADP] cytoplasmic|Homo sapiens (9606)
>1TCV_1|Chains A, B, C|purine-nucleoside phosphorylase|Schistosoma mansoni (6183)
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 \(w=f(c)\)
1ZBB , Knot 50 347 0.28 8 16 57 
ACTTACATGCACAGGATGTAACCTGCAGATACTACCAAAAGTGTATTTGGAAACTGCTCCATCAAAAGGCATGTTCAGCTGGATTCCAGCTGAACATGCCTTTTGATGGAGCAGTTTCCAAATACACTTTTGGTAGTATCTGCAGGTGATTCTCCAGGGCGGCCAGTACTTACATGCACAGGATGTAACCTGCAGATACTACCAAAAGTGTATTTGGAAACTGCTCCATCAAAAGGCATGTTCAGCTGGATTCCAGCTGAACATGCCTTTTGATGGAGCAGTTTCCAAATACACTTTTGGTAGTATCTGCAGGTGATTCTCCAGACTTACATGCGCATGTAAGTGCA
5LGE , Knot 178 425 0.84 40 234 407 
MSKKISGGSVVEMQGDEMTRIIWELIKEKLIFPYVELDLHSYDLGIENRDATNDQVTKDAAEAIKKHNVGVKCATITPDEKRVEEFKLKQMWKSPNGTIRNILGGTVFREAIICKNIPRLVSGWVKPIIIGHHAYGDQYRATDFVVPGPGKVEITYTPSDGTQKVTYLVHNFEEGGGVAMGMYNQDKSIEDFAHSSFQMALSKGWPLYLSTKNTILKKYDGRFKDIFQEIYDKQYKSQFEAQKIWYEHRLIDDMVAQAMKSEGGFIWACKNYDGDVQSDSVAQGYGSLGMMTSVLVCPDGKTVEAEAAHGTVTRHYRMYQKGQETSTNPIASIFAWTRGLAHRAKLDNNKELAFFANALEEVSIETIEAGFMTKDLAACIKGLPNVQRSDYLNTFEFMDKLGENLKIKLAQAKLSLEHHHHHHHH
1TCV , Knot 132 287 0.86 40 184 276 
MHESVTANIENVKKVAHHIQKLTSIVPEIGIICGSGLGKLADGVKDKITIPYTKIPNFPQTSVVGHSGNLIFGTLSGRKVVVMQGRFHMYEGYSNDTVALPIRVMKLLGVKILMVSNAAGGLNRSLKLGDFVILKDHIYLPGLGLNNILVGPNQEAFGTRFPALSNAYDRDLRKLAVQVAEENGFGNLVHQGVYVMNGGPCYETPAECTMLLNMGCDVVGMSTIPEVVIARHCGIQVFAVSLVTNISVLDVESDLKPNHEEVLATGAQRAELMQSWFEKIIEKLPKD

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 \(c\).

\(|P_{f(1ZBB_1)}(2) \setminus P_{f(5LGE_1)}(2)|=9\), \(|P_{f(5LGE_1)}(2) \setminus P_{f(1ZBB_1)}(2)|=227\). 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 1:10001010101011110101100010111010010011111010100011111001000010011111101010001100111000011001110101000000110111101100000111010100000110110100010111011000000111101100110100010101010111101011000101110100100111110101000111110010000100111111010100011001110000110011101010000001101111011000001110101000001101101000101110110000001110001010101010101110101
Pair \(Z_2\) Length of longest common subsequence
1ZBB_1,5LGE_1 236 3
1ZBB_1,1TCV_1 178 4
5LGE_1,1TCV_1 152 3

Newick tree

 
[
	1ZBB_1:11.41,
	[
		1TCV_1:76,5LGE_1:76
	]:36.41
]

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 \((0.85)(0.8)(\frac{772 }{\log_{20} 772}-\frac{347}{\log_{20}347})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 1ZBB_1 5LGE_1 174 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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