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Parikh vectors
1YGP_1 4JGS_1 1LKY_1 Letter Amino acid
52 10 4 A Alanine
43 7 7 R Arginine
33 6 4 Q Glutamine
67 9 5 E Glutamic acid
61 6 4 K Lycine
16 1 1 M Methionine
50 11 6 S Serine
5 2 0 C Cysteine
46 9 2 G Glycine
60 1 4 I Isoleucine
42 4 3 F Phenylalanine
58 7 1 V Valine
53 2 3 N Asparagine
18 2 3 H Histidine
40 5 2 T Threonine
17 0 3 W Tryptophan
36 1 3 Y Tyrosine
59 4 5 D Aspartic acid
86 18 13 L Leucine
37 0 4 P Proline 

1YGP_1|Chains A, B|YEAST GLYCOGEN PHOSPHORYLASE|Saccharomyces cerevisiae (4932)
>4JGS_1|Chains A, B, C, D, E, F, G, H, I|MLV-related proviral Env polyprotein|Mus musculus (10090)
>1LKY_1|Chains A, C, E|TRANSCRIPTION FACTOR ETV6|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 \(w=f(c)\)
1YGP , Knot 331 879 0.85 40 310 784 
PRLTRRLTGFLPQEIKSIDTMIPLLSRALWNKHQVKKFNKAEDFQDRFIDHVETTLARSLYNCDDMVAYEAASMSIRDNLVIDWNKTQQKFTTRDPKRVYYLSLEFLMGRALDNALINMKIEDPEDPAASKGKPREMIKGALDELGFKLEDVLDQEPDAGLGNGGLGRLAACFVDSMATEGIPAWGYGLRYEYGIFAQKIIDGYQVETPDYWLNSGNPWEIERNEVQIPVTFYGYVDRPEGGKTTLSASQWIGGERVLAVAYDFPVPGFKTSNVNNLRLWQARPTTEFDLNKFNNGDYKNSVAQQQRAESITAVLYPNDNFAQGKELRLKQQYFWCAASLHDILRRFKKSKRPWTEFPDQVAIQLNDTHPTLAIVELQRVLVDLEKLDWHEAWDIVTKTFAYTNHTVMQEALEKWPRRLFGHLLPRHLEIIYDINWFFLEDVAKKFPKDVDLLSRISIIEENSPERQIRMAFLAIVGSHKVNGVVELHSELIKTTIFKDFIKFYGPSKFVNVTNGITPRRWLKQANPSLAKLISETLNDPTEEYLLDMAKLTQLEKYVEDKEFLKKWNQVKLNNKIRLVDLIKKENDGVDIINREYLDDTLFDMQVKRIHEYKRQQLNVFGIIYRYLAMKNMLKNGASIEEVARKYPRKVSIFGGKSAPGYYMAKLIIKLINCVADIVNNDESIEHLLKVVFVADYNVSKAEIIIPASDLSEHISTAGTEASGTSNMKFVMNGGLIIGTVDGANVEITREIGEDNVFLFGNLSENVEELRYNHQYHPQDLPSSLDSVLSYIESGQFSPENPNEFKPLVDSIKYHGDYYLVSDDFESYLATHELVDQEFHNQRSEWLKKSVLSLANVGFFSSDRCIEEYSDTIWNVEPVT
4JGS , Knot 52 105 0.76 36 81 99 
SSGLVGSGTALVATKQFEQLQAAIHTDLGALEKSVSALEKSLTSLSEVVLQNRRGLDLLFLKEGGLCAALKEECSFYADHTGVVRDSMAKLRERLNQRQRLFESG
1LKY , Knot 45 77 0.84 38 67 74 
SIRLPAHLRLQPIYWSRDDVAQWLKWAENEFSLRPIDSNTFEMNGKALLLLTKEDFRYRSPHSGDRLYELLQHILKQ

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 \(c\).

\(|P_{f(1YGP_1)}(2) \setminus P_{f(4JGS_1)}(2)|=239\), \(|P_{f(4JGS_1)}(2) \setminus P_{f(1YGP_1)}(2)|=10\). 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 1:101000101111001001001111100111000010010010010001100100011001000001110011010100011101000000100001001001010111101100111010100100111001010011011100111010011000101111011110111011001100111111011000011110011010010010011001011010000101110101010010110001010011110011111001111110000100101101010001010010010000011000010010111010001101001010000110110100110010000011001100111010000101111010011101001010011011000110000011001100110011101110010110010111100110011001011001011000010001011111111000101110100011000110011010110011010011010011001010110110001001000011011010010001000011001001010001011011000001101100001000110101001000000010111110001110011001101001100010010111100111001101110110011011000001001101111100010010111110010001001100101000101110111111010110101000110001111101000100100000001001100100110010010101001001011100100010001100010001100011000100000011000110110111100000100000011010110
Pair \(Z_2\) Length of longest common subsequence
1YGP_1,4JGS_1 249 4
1YGP_1,1LKY_1 261 3
4JGS_1,1LKY_1 116 2

Newick tree

 
[
	1YGP_1:14.40,
	[
		4JGS_1:58,1LKY_1:58
	]:85.40
]

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 \((0.85)(0.8)(\frac{984 }{\log_{20} 984}-\frac{105}{\log_{20}105})=244.\)
Status Protein1 Protein2 d d1/2
Query variables 1YGP_1 4JGS_1 312 172.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]