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Parikh vectors
4DOH_1 4CGZ_1 5MNU_1 Letter Amino acid
3 23 8 P Proline
9 33 11 T Threonine
10 36 18 A Alanine
11 33 8 R Arginine
7 40 6 D Aspartic acid
11 36 23 E Glutamic acid
10 38 16 I Isoleucine
23 65 18 L Leucine
1 4 7 W Tryptophan
4 25 6 Y Tyrosine
7 30 14 N Asparagine
6 19 2 C Cysteine
9 50 22 S Serine
6 22 2 H Histidine
11 56 21 K Lycine
3 19 5 M Methionine
5 44 20 V Valine
7 28 10 Q Glutamine
6 36 18 G Glycine
4 28 12 F Phenylalanine 

4DOH_1|Chains A, D[auth C]|Interleukin-20|Homo sapiens (9606)
>4CGZ_1|Chain A|BLOOM'S SYNDROME HELICASE|HOMO SAPIENS (9606)
>5MNU_1|Chains A, B|Beta-lactamase OXA-10|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
4DOH , Knot 77 153 0.84 40 122 148 
ALKTLNLGSCVIATNLQEIRNGFSEIRGSVQAKDGNIDIRILRRTESLQDTKPANRCCLLRHLLRLYLDRVFKNYQTPDHYTLRKISSLANSFLTIKKDLRLCHAHMTCHCGEEAMKKYSQILSHFEKLEPQAAVVKALGELDILLQWMEETE
4CGZ , Knot 264 665 0.86 40 296 622 
SMSRNLKHERFQSLSFPHTKEMMKIFHKKFGLHNFRTNQLEAINAALLGEDCFILMPTGGGKSLCYQLPACVSPGVTVVISPLRSLIVDQVQKLTSLDIPATYLTGDKTDSEATNIYLQLSKKDPIIKLLYVTPEKICASNRLISTLENLYERKLLARFVIDEAHCVSQWGHDFRQDYKRMNMLRQKFPSVPVMALTATANPRVQKDILTQLKILRPQVFSMSFNRHNLKYYVLPKKPKKVAFDCLEWIRKHHPYDSGIIYCLSRRECDTMADTLQRDGLAALAYHAGLSDSARDEVQQKWINQDGCQVICATIAFGMGIDKPDVRFVIHASLPKSVEGYYQESGRAGRDGEISHCLLFYTYHDVTRLKRLIMMEKDGNHHTRETHFNNLYSMVHYCENITECRRIQLLAYFGENGFNPDFCKKHPDVSCDNCCKTKDYKTRDVTDDVKSIVRFVQEHSSSQGMRNIKHVGPSGRFTMNMLVDIFLGSKSAKIQSGIFGKGSAYSRHNAERLFKKLILDKILDEDLYINANDQAIAYVMLGNKAQTVLNGNLKVDFMETENSSSVKKQKALVAKVSQREEMVKKCLGELTEVCKSLGKVFGVHYFNIFNTVTLKKLAESLSSDPEVLLQIDGVTEDKLEKYGAEVISVLQKYSEWTSPAEDSSPG
5MNU , Knot 115 247 0.85 40 168 239 
MGSITENTSWNKEFSAEAVNGVFVLCKSSSKSCATNDLARASKEYLPASTFKIPNAIIGLETGVIKNEHQVFKWDGKPRAMKQWERDLTLRGAIQVSAVPVFQQIAREVGEVRMQKYLKKFSYGNQNISGGIDKFWLEGQLRISAVNQVEFLESLYLNKLSASKENQLIVKEALVTEAAPEYLVHSKTGFSGVGTESNPGVAWWVGWVEKETEVYFFAFNMDIDNESKLPLRKSIPTKIMESEGIIG

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 \(c\).

\(|P_{f(4DOH_1)}(2) \setminus P_{f(4CGZ_1)}(2)|=17\), \(|P_{f(4CGZ_1)}(2) \setminus P_{f(4DOH_1)}(2)|=191\). 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 1:110010110011100100100110010101010010101011000001000011000011001101010011000001000010010011001101000101001010000100110000011001001010111101110101110110000
Pair \(Z_2\) Length of longest common subsequence
4DOH_1,4CGZ_1 208 4
4DOH_1,5MNU_1 168 3
4CGZ_1,5MNU_1 192 4

Newick tree

 
[
	4CGZ_1:10.89,
	[
		4DOH_1:84,5MNU_1:84
	]:20.89
]

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 \((0.85)(0.8)(\frac{818 }{\log_{20} 818}-\frac{153}{\log_{20}153})=186.\)
Status Protein1 Protein2 d d1/2
Query variables 4DOH_1 4CGZ_1 238 144
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]