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Parikh vectors
7JIS_1 2LAL_1 3DLG_1 Letter Amino acid
26 0 6 M Methionine
29 7 22 Y Tyrosine
60 16 40 V Valine
51 7 37 P Proline
29 0 2 C Cysteine
51 6 37 Q Glutamine
30 2 8 H Histidine
66 14 19 S Serine
67 13 26 A Alanine
55 4 18 R Arginine
78 5 47 E Glutamic acid
37 10 41 I Isoleucine
142 8 49 L Leucine
65 11 58 K Lycine
46 13 13 F Phenylalanine
41 24 40 T Threonine
36 15 19 N Asparagine
44 11 25 D Aspartic acid
48 12 34 G Glycine
17 3 19 W Tryptophan 

7JIS_1|Chain A|Phosphatidylinositol 4,5-bisphosphate 3-kinase catalytic subunit delta isoform|Homo sapiens (9606)
>2LAL_1|Chains A, C|LENTIL LECTIN (ALPHA CHAIN)|Lens culinaris (3864)
>3DLG_1|Chain A|REVERSE TRANSCRIPTASE/RIBONUCLEASE H|HIV-1 M:B_HXB2R (11706)
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 \(w=f(c)\)
7JIS , Knot 380 1018 0.86 40 323 906 
NQSVVVDFLLPTGVYLNFPVSRNANLSTIKQLLWHRAQYEPLFHMLSGPEAYVFTCINQTAEQQELEDEQRRLCDVQPFLPVLRLVAREGDRVKKLINSQISLLIGKGLHEFDSLCDPEVNDFRAKMCQFCEEAAARRQQLGWEAWLQYSFPLQLEPSAQTWGPGTLRLPNRALLVNVKFEGSEESFTFQVSTKDVPLALMACALRKKATVFRQPLVEQPEDYTLQVNGRHEYLYGSYPLCQFQYICSCLHSGLTPHLTMVHSSSILAMRDEQSNPAPQVQKPRAKPPPIPAKKPSSVSLWSLEQPFRIELIQGSKVNADERMKLVVQAGLFHGNEMLCKTVSSSEVSVCSEPVWKQRLEFDINICDLPRMARLCFALYAVIEKAKKARSTKKKSKKADCPIAWANLMLFDYKDQLKTGERCLYMWPSVPDEKGELLNPTGTVRSNPNTDSAAALLICLPEVAPHPVYYPALEKILELGRHSECVHVTEEEQLQLREILERRGSGELYEHEKDLVWKLRHEVQEHFPEALARLLLVTKWNKHEDVAQMLYLLCSWPELPVLSALELLDFSFPDCHVGSFAIKSLRKLTDDELFQYLLQLVQVLKYESYLDCELTKFLLDRALANRKIGHFLFWHLRSEMHVPSVALRFGLILEAYCRGSTHHMKVLMKQGEALSKLKALNDFVKLSSQKTPKPQTKELMHLCMRQEAYLEALSHLQSPLDPSTLLAEVCVEQCTFMDSKMKPLWIMYSNEEAGSGGSVGIIFKNGDDLRQDMLTLQMIQLMDVLWKQEGLDLRMTPYGCLPTGDRTGLIEVVLRSDTIANIQLNKSNMAATAAFNKDALLNWLKSKNPGEALDRAIEEFTLSCAGYCVATYVLGIGDRHSDNIMIRESGQLFHIDFGHFLGNFKTKFGINRERVPFILTYDFVHVIQQGKTNNSEKFERFRGYCERAYTILRRHGLLFLHLFALMRAAGLPELSCSKDIQYLKDSLALGKTEEEALKHFRVKFNEALRESWKTKVNWL
2LAL , Knot 81 181 0.77 36 124 174 
TETTSFSITKFSPDQQNLIFQGDGYTTKGKLTLTKAVKSTVGRALYSTPIHIWDRDTGNVANFVTSFTFVIDAPSSYNVADGFTFFIAPVDTKPQTGGGYLGVFNSKEYDKTSQTVAVEFDTFYNAAWDPSNKERHIGIDVNSIKSVNTKSWNLQNGERANVVIAFNAATNVLTVTLTYPN
3DLG , Knot 223 560 0.84 40 255 517 
PISPIETVPVKLKPGMDGPKVKQWPLTEEKIKALVEICTEMEKEGKISKIGPENPYNTPVFAIKKKDSTKWRKLVDFRELNKRTQDFWEVQLGIPHPAGLKKKKSVTVLDVGDAYFSVPLDEDFRKYTAFTIPSINNETPGIRYQYNVLPQGWKGSPAIFQSSMTKILEPFRKQNPDIVIYQYMDDLYVGSDLEIGQHRTKIEELRQHLLRWGLTTPDKKHQKEPPFLWMGYELHPDKWTVQPIVLPEKDSWTVNDIQKLVGKLNWASQIYPGIKVRQLCKLLRGTKALTEVIPLTEEAELELAENREILKEPVHGVYYDPSKDLIAEIQKQGQGQWTYQIYQEPFKNLKTGKYARMRGAHTNDVKQLTEAVQKITTESIVIWGKTPKFKLPIQKETWETWWTEYWQATWIPEWEFVNTPPLVKLWYQLEKEPIVGAETFYVDGAANRETKLGKAGYVTNRGRQKVVTLTDTTNQKTELQAIYLALQDSGLEVNIVTDSQYALGIIQAQPDQSESELVNQIIEQLIKKEKVYLAWVPAHKGIGGNEQVDKLVSAGIRKVL

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 \(c\).

\(|P_{f(7JIS_1)}(2) \setminus P_{f(2LAL_1)}(2)|=213\), \(|P_{f(2LAL_1)}(2) \setminus P_{f(7JIS_1)}(2)|=14\). 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 1:0001110111101101011100010100100111001000111011011010110010001000010000001001011111101110010010011000101111011001001001010010101001000111000011101110001110101010011110101100111101010100001010100001111111011000101100111001000010101000010100110010010001001101010110000111100000011101001010111111001001011010011010110100101000101110111101001100010000101000111000101010100110110101110111001001000000000100111110111100000100100010111011000101101010100010000111111011011101100111001101100000101000001010011000101010000001110100010001101110111100100000110110110011011110110110101100011011100100100001100110110110000010001001110011100011011110100010110111011111010001000010111001011001011001101000001010000110101000101011001001101001110101000011000101111100000110110111110010010001101011011011100011010101010110100011101110000110101000011101110001110110000110110011001010011001100111110000001110001011010110111010001110000111110001101100100000001001010000100110001111101111101111101000001001000111100000110010101001100010001011
Pair \(Z_2\) Length of longest common subsequence
7JIS_1,2LAL_1 227 4
7JIS_1,3DLG_1 146 4
2LAL_1,3DLG_1 193 4

Newick tree

 
[
	2LAL_1:11.10,
	[
		7JIS_1:73,3DLG_1:73
	]:41.10
]

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 \((0.85)(0.8)(\frac{1199 }{\log_{20} 1199}-\frac{181}{\log_{20}181})=273.\)
Status Protein1 Protein2 d d1/2
Query variables 7JIS_1 2LAL_1 358 208.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]