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Parikh vectors
8GEO_1 3CAZ_1 7TYQ_1 Letter Amino acid
24 13 16 R Arginine
37 14 5 N Asparagine
9 2 5 C Cysteine
41 13 9 T Threonine
35 11 10 Y Tyrosine
44 18 20 V Valine
31 7 13 F Phenylalanine
30 31 13 A Alanine
18 18 12 Q Glutamine
43 41 19 E Glutamic acid
51 5 17 G Glycine
12 8 8 H Histidine
43 15 24 L Leucine
40 30 8 K Lycine
10 4 3 W Tryptophan
44 14 9 D Aspartic acid
27 16 5 I Isoleucine
17 3 4 M Methionine
26 8 12 P Proline
32 23 19 S Serine 

8GEO_1|Chain A|Beta-glucuronidase|Lachnospira eligens (39485)
>3CAZ_1|Chains A, B|BAR protein|Galdieria sulphuraria (130081)
>7TYQ_1|Chains A, B|Transcriptional enhancer factor TEF-4|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 \(w=f(c)\)
8GEO , Knot 242 614 0.84 40 287 581 
SNAMLYPVLTQSRLLSDLSGVWNFKLDNGKGFEEKWYEKPLKDADTMPVPASYNDLKEGTDFRDHYGWVFYQRNISVPEYVKSQRIVLRCAAVTHYAMIYLNGKLICEHKGGFLPFEVELNDDLQDGDNLLTIAVNNVIDYTTLPVGGKANMMSGMMGGMGAGASDKPQNNPNFDFFNYCGITRPVKIYTTPETYINDITVTADIDFTKEEPSAVLNYNVEIKGKDYNNITCKVELFDEEGTKLSETEGSEGTFEISNVRLWQPLNAYLYKIKVTAGQDVYTLPYGVRSVRVDGTKFLINEKPFYFKGYGKHEDTFPNGRGINLPMNTKDISIMKWQHANSFRTSHYPYSEEMMRLCDEEGIVVIDETTAVGVNLQFGGGANFGGERIGTFDKEHGVQTQEHHKDVIRDLISRDKNHACVVMWSIANEPDSAAEGAYDYFKPLYDLARELDPQKRPCTLVSVQGTTADTDCSSQLSDVICLNRYYGWYFGGPDLEVSETGLRKELSDWGKLGKPVMFTEYGADTVSGLHDTTSVMYTEEYQVEYYEMNNKVFDEFDFVVGEQAWNFADFATSQSLLRVQGNKKGLFTRDRKPKMVAHYFRNRWSAIPEFGYKTK
3CAZ , Knot 127 294 0.81 40 175 275 
AIAERLSPVFATSKSSNMEVEHWMTSLKEVEKVLDSVEPKLTSSGTKWRVVAQHIKDICSDLNQIFNKEDPRYEVVQAGASAAHDFDVRYLDIHKHGREIARLLEKIQKYRQEIEEIKKEYKETDKYRERYDHYKVKLDNLEKKNKDQERIERNQQKFKDAEAAYSSVCADLIQKMETVWKKHVSIFAEAASAVWSTQLQYAKALEAAANPIVPYLQQEEQEEEEEEEAASGTQGRVAEKAYHTNEEKVENSFQKNSAIPPSFRTASEQLTEQSKNINNAKNTSETSPAEDIEA
7TYQ , Knot 108 231 0.84 40 164 227 
AWQARGLGTARLQLVEFSAFVEPPDAVDSYQRHLFVHISQHCPSPGAPPLESVDVRQIYDKFPEKKGGLRELYDRGPPHAFFLVKFWADLNWGPSGEEAGAGGSISSGGFYGVSSQYESLEHMTLTCSSKVCSFGKQVVEKVETERAQLEDGRFVYRLLRSPMCEYLVNFLHKLRQLPERYMMNSVLENFTILQVVTNRDTQELLLCTAYVFEVSTSERGAQHHIYRLVRD

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 \(c\).

\(|P_{f(8GEO_1)}(2) \setminus P_{f(3CAZ_1)}(2)|=148\), \(|P_{f(3CAZ_1)}(2) \setminus P_{f(8GEO_1)}(2)|=36\). 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 1:00111011100001100101110101001011000100011001001111100001001001000011110000101100100001110011100011101010110000111111010100010010011011100110000111110101101111111111000100010101100011001101000100010010101010100001011100010101000001000101100010010000100101010010110110101001010110010011011001010100111000110101010000011010110111000010110100100100000100001101000011111000011110101111101110011010000110000000011001100000010111101100100110110001011001100101000100110101001000000010011010000110111101010001100010011011011110001100101100000110000001000010001100101111001101101100001101010001110000010111001000101110110000
Pair \(Z_2\) Length of longest common subsequence
8GEO_1,3CAZ_1 184 4
8GEO_1,7TYQ_1 195 3
3CAZ_1,7TYQ_1 181 4

Newick tree

 
[
	8GEO_1:96.17,
	[
		3CAZ_1:90.5,7TYQ_1:90.5
	]:5.67
]

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 \((0.85)(0.8)(\frac{908 }{\log_{20} 908}-\frac{294}{\log_{20}294})=166.\)
Status Protein1 Protein2 d d1/2
Query variables 8GEO_1 3CAZ_1 217 157
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]