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Parikh vectors
9GGP_1 6TNR_1 5BNF_1 Letter Amino acid
23 50 14 G Glycine
10 31 2 M Methionine
8 54 12 R Arginine
19 34 13 N Asparagine
18 41 10 Q Glutamine
1 23 10 C Cysteine
31 75 12 E Glutamic acid
17 42 7 P Proline
34 65 19 K Lycine
27 42 12 F Phenylalanine
31 36 16 T Threonine
24 55 14 V Valine
24 63 9 A Alanine
24 39 14 D Aspartic acid
19 30 5 H Histidine
2 16 5 W Tryptophan
6 27 6 Y Tyrosine
19 34 12 I Isoleucine
45 122 20 L Leucine
22 61 20 S Serine 

9GGP_1|Chains A, B|Alpha-1-antitrypsin|Homo sapiens (9606)
>6TNR_1|Chain A|Phosphatidylinositol 4,5-bisphosphate 3-kinase catalytic subunit delta isoform|Mus musculus (10090)
>5BNF_1|Chains A, B|Uncharacterized protein|Sus scrofa (9823)
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 \(w=f(c)\)
9GGP , Knot 169 404 0.83 40 223 374 
MRGSHHHHHHTDPQGDAAQKTDTSHHDQDHPTFNKITPNLAEFAFSLYRQLAHQSNSTNIFFSPVSIATAFAMLSLGTKADTHDEILEGLNFNLTEIPEAQIHEGFQELLRTLNQPDSQLQLTTGNGLFLSEGLKLVDKFLEDVKKLYHSEAFTVNFGDTEEAKKQINDYVEKGTQGKIVDLVKELDRDTVFALVNYIFFKGKWERPFEVKDTEEEDFHVDQVTTVKVPMMKRLGMFNIQHCKKLSSWVLLMKYLGNATAIFFLPDEGKLQHLENELTHDIITKFLENEDRRSASLHLPKLSITGTYDLKSVLGQLGITKVFSNGADLSGVTEEAPLKLSKAVHKAVLTIDEKGTEAAGAMFLEAIPMSIPPEVKFNKPFVFLMIEQNTKSPLFMGKVVNPTQK
6TNR , Knot 354 940 0.86 40 322 842 
GGDRVKKLINSQISLLIGKGLHEFDSLRDPEVNDFRTKMRQFCEEAAAHRQQLGWVEWLQYSFPLQLEPSARGWRAGLLRVSNRALLVNVKFEGSEESFTFQVSTKDMPLALMACALRKKATVFRQPLVEQPEEYALQVNGRHEYLYGNYPLCHFQYICSCLHSGLTPHLTMVHSSSILAMRDEQSNPAPQVQKPRAKPPPIPAKKPSSVSLWSLEQPFSIELIEGRKVNADERMKLVVQAGLFHGNEMLCKTVSSSEVNVCSEPVWKQRLEFDISVCDLPRMARLCFALYAVVEKAKKARSTKKKSKKADCPIAWANLMLFDYKDQLKTGERCLYMWPSVPDEKGELLNPAGTVRGNPNTESAAALVIYLPEVAPHPVYFPALEKILELGRHGERGRITEEEQLQLREILERRGSGELYEHEKDLVWKMRHEVQEHFPEALARLLLVTKWNKHEDVAQMLYLLCSWPELPVLSALELLDFSFPDCYVGSFAIKSLRKLTDDELFQYLLQLVQVLKYESYLDCELTKFLLGRALANRKIGHFLFWHLRSEMHVPSVALRFGLIMEAYCRGSTHHMKVLMKQGEALSKLKALNDFVKVSSQKTTKPQTKEMMHMCMRQETYMEALSHLQSPLDPSTLLEEVCVEQCTFMDSKMKPLWIMYSSEEAGSAGNVGIIFKNGDDLRQDMLTLQMIQLMDVLWKQEGLDLRMTPYGCLPTGDRTGLIEVVLHSDTIANIQLNKSNMAATAAFNKDALLNWLKSKNPGEALDRAIEEFTLSCAGYCVATYVLGIGDRHSDNIMIRESGQLFHIDFGHFLGNFKTKFGINRERVPFILTYDFVHVIQQGKTNNSEKFERFRGYCERAYTILRRHGLLFLHLFALMRAAGLPELSCSKDIQYLKDSLALGKTEEEALKHFRVKFNEALRESWKTKVNWLAHNVSKDNRQ
5BNF , Knot 106 232 0.83 40 158 226 
WNGKGSTVDFQEIILRRCYTYIRVVQPELGDRDCQKIKKAFTDAFISKDPCSAREEDYDLLMKLGHQTVPCDKTVFWSKTKELAHQYTKTQKGLFTLENTLLGYIADDLSWCGKVGSSEINLESCPDRRNCNSNFVSVFWNLLSKRFAENACGMVQVFLNGSISNAFDKTSTFGRVEVHSLQPSKVHTLKAWVIHDSGKTPRDTCSGSSINELQLILRGKNIKFTCQENYRP

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 \(c\).

\(|P_{f(9GGP_1)}(2) \setminus P_{f(6TNR_1)}(2)|=33\), \(|P_{f(6TNR_1)}(2) \setminus P_{f(9GGP_1)}(2)|=132\). 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 1:10100000000010101100000000000010100101011011101000110000000111011011011111011001000001101101010011010100110011001001000101001011110011011001100100100001101011000010001000100100101101100100001111100111010100110100000001010010010111100111101000001001111100110101111110010100100010001100110000000101011010101000100111011100110011010110001110100110011101000100111111101111011101010011111110000001111101101000
Pair \(Z_2\) Length of longest common subsequence
9GGP_1,6TNR_1 165 4
9GGP_1,5BNF_1 155 4
6TNR_1,5BNF_1 208 4

Newick tree

 
[
	6TNR_1:98.72,
	[
		9GGP_1:77.5,5BNF_1:77.5
	]:21.22
]

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 \((0.85)(0.8)(\frac{1344 }{\log_{20} 1344}-\frac{404}{\log_{20}404})=242.\)
Status Protein1 Protein2 d d1/2
Query variables 9GGP_1 6TNR_1 311 219
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]