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Parikh vectors
1VEU_1 4YQA_1 9JUJ_1 Letter Amino acid
6 14 63 D Aspartic acid
1 3 14 C Cysteine
7 22 72 E Glutamic acid
16 30 128 L Leucine
6 7 59 F Phenylalanine
13 9 117 S Serine
4 5 39 Y Tyrosine
5 4 43 N Asparagine
3 11 52 Q Glutamine
2 8 26 H Histidine
14 16 102 V Valine
4 16 67 R Arginine
7 13 83 I Isoleucine
9 14 71 K Lycine
1 7 33 M Methionine
6 14 42 P Proline
4 13 59 T Threonine
6 25 98 G Glycine
0 5 17 W Tryptophan
10 18 142 A Alanine 

1VEU_1|Chain A|Mitogen-activated protein kinase kinase 1 interacting protein 1|Mus musculus (10090)
>4YQA_1|Chain A|tRNA (guanine-N(1)-)-methyltransferase|Haemophilus influenzae (strain ATCC 51907 / DSM 11121 / KW20 / Rd) (71421)
>9JUJ_1|Chain A|ABC transporter B family member 1|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
1VEU , Knot 64 124 0.83 38 95 120 
MADDLKRFLYKKLPSVEGLHAIVVSDRDGVPVIKVANDSAPEHALRPGFLSTFALATDQGSKLGLSKNKSIICYYNTYQVVQFNRLPLVVSFIASSSANTGLIVSLEKELAPLFEELIKVVEVS
4YQA , Knot 116 254 0.84 40 175 245 
GLVPRGSHMWIGVISLFPEMFKAITEFGVTGRAVKHNLLKVECWNPRDFTFDKHKTVDDRPYGGGPGMLMMVQPLRDAIHTAKAAAGEGAKVIYLSPQGRKLDQGGVTELAQNQKLILVCGRYEGIDERLIQTEIDEEWSIGDYVLTGGELPAMTLIDAVARFIPGVLGKQASAEEDSFADGLLDCPHYTRPEVLEGLTVPPVLMSGHHEEIRKWRLKQSLQRTWLRRPELLEGLALTDEQRKLLKEAQAEHNS
9JUJ , Knot 452 1327 0.81 40 335 1063 
MDYKDDDDKWSHPQFEKGGGGSGGSAWSHPQFEKEFKGLVDMDNDGGAPPPPPTLVVEEPKKAEIRGVAFKELFRFADGLDYVLMGIGSVGAFVHGCSLPLFLRFFADLVNSFGSNSNNVEKMMEEVLKYALYFLVVGAAIWASSWAEISCWMWSGERQTTKMRIKYLEAALNQDIQFFDTEVRTSDVVFAINTDAVMVQDAISEKLGNFIHYMATFVSGFIVGFTAVWQLALVTLAVVPLIAVIGGIHTTTLSKLSNKSQESLSQAGNIVEQTVVQIRVVMAFVGESRASQAYSSALKIAQKLGYKTGLAKGMGLGATYFVVFCCYALLLWYGGYLVRHHLTNGGLAIATMFAVMIGGLALGQSAPSMAAFAKAKVAAAKIFRIIDHKPTIERNSESGVELDSVTGLVELKNVDFSYPSRPDVKILNNFCLSVPAGKTIALVGSSGSGKSTVVSLIERFYDPNSGQVLLDGQDLKTLKLRWLRQQIGLVSQEPALFATSIKENILLGRPDADQVEIEEAARVANAHSFIIKLPDGFDTQVGERGLQLSGGQKQRIAIARAMLKNPAILLLDEATSALDSESEKLVQEALDRFMIGRTTLIIAHRLSTIRKADLVAVLQQGSVSEIGTHDELFSKGENGVYAKLIKMQEAAHETAMSNARKSSARPSSARNSVSSPIMTRNSSYGRSPYSRRLSDFSTSDFSLSIDASSYPNYRNEKLAFKDQANSFWRLAKMNSPEWKYALLGSVGSVICGSLSAFFAYVLSAVLSVYYNPDHEYMIKQIDKYCYLLIGLSSAALVFNTLQHSFWDIVGENLTKRVREKMLSAVLKNEMAWFDQEENESARIAARLALDANNVRSAIGDRISVIVQNTALMLVACTAGFVLQWRLALVLVAVFPVVVAATVLQKMFMTGFSGDLEAAHAKGTQLAGEAIANVRTVAAFNSEAKIVRLYTANLEPPLKRCFWKGQIAGSGYGVAQFCLYASYALGLWYASWLVKHGISDFSKTIRVFMVLMVSANGAAETLTLAPDFIKGGQAMRSVFELLDRKTEIEPDDPDTTPVPDRLRGEVELKHIDFSYPSRPDIQIFRDLSLRARAGKTLALVGPSGCGKSSVISLIQRFYEPSSGRVMIDGKDIRKYNLKAIRKHIAIVPQEPCLFGTTIYENIAYGHECATEAEIIQAATLASAHKFISALPEGYKTYVGERGVQLSGGQKQRIAIARALVRKAEIMLLDEATSALDAESERSVQEALDQACSGRTSIVVAHRLSTIRNAHVIAVIDDGKVAEQGSHSHLLKNHPDGIYARMIQLQRFTHTQVIGMTSGSSSRVKEDDA

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 \(c\).

\(|P_{f(1VEU_1)}(2) \setminus P_{f(4YQA_1)}(2)|=42\), \(|P_{f(4YQA_1)}(2) \setminus P_{f(1VEU_1)}(2)|=122\). 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 1:1100100110001101011011110000111110110001100110111100111100010011100000110000000110100111110111000100111101000111110011011010
Pair \(Z_2\) Length of longest common subsequence
1VEU_1,4YQA_1 164 3
1VEU_1,9JUJ_1 250 3
4YQA_1,9JUJ_1 192 5

Newick tree

 
[
	9JUJ_1:11.66,
	[
		1VEU_1:82,4YQA_1:82
	]:37.66
]

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 \((0.85)(0.8)(\frac{378 }{\log_{20} 378}-\frac{124}{\log_{20}124})=77.3\)
Status Protein1 Protein2 d d1/2
Query variables 1VEU_1 4YQA_1 99 72
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]