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Parikh vectors
4LAQ_1 2BJR_1 5OOA_1 Letter Amino acid
6 17 7 R Arginine
4 16 5 N Asparagine
8 21 4 D Aspartic acid
4 10 0 C Cysteine
12 10 4 Q Glutamine
8 8 5 H Histidine
50 16 12 S Serine
9 22 4 I Isoleucine
15 25 4 L Leucine
19 10 9 T Threonine
17 15 8 Y Tyrosine
12 21 16 V Valine
11 26 6 A Alanine
8 23 11 E Glutamic acid
11 29 6 P Proline
6 13 0 W Tryptophan
28 43 8 G Glycine
10 24 3 K Lycine
4 8 3 M Methionine
7 11 9 F Phenylalanine 

4LAQ_1|Chain A[auth H]|Single chain antibody fragment scFv6H4|Mus musculus (10090)
>2BJR_1|Chains A, B|MFP2B|ASCARIS SUUM (6253)
>5OOA_1|Chain A|Putative cupin_2 domain-containing isomerase|Streptomyces coeruleorubidus (116188)
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 \(w=f(c)\)
4LAQ , Knot 105 249 0.77 40 148 215 
EVQLQESGPSLVKPSQTLSLTCSVTGDSVTSGYWSWIRQFPGNKLDYMGYISYRGSTYYNPSLKSRISITRDTSKNQVYLQLKSVSSEDTATYYCSYFDSDDYAMEYWGQGTSVTVSGGGGSGGGGSGGGGSQIVLTQSPAIMSASPGEKVTLTCSASSSVSSSHLYWYQQKPGSSPKLWIYSTSNLASGVPARFSGSGSGTSYSLTISSMEAEDAASYFCHQWSSFPFTFGSGTKLEIKRAPHHHHHH
2BJR , Knot 156 368 0.83 40 221 348 
MPNPPAKEDTWAFGPIGSPFPDNPVKALGQQNMYVALWYKNGRPMHGRAWNNGGVIECSFPYNKSELTGVKDLGGQIQVLQYKGNHLSLGYWYNWIKYSDRFDKMDKGAEMLRCGDSFPILWSERPGGALLGYADNKTEIARFSHDGKVDEVSGSALANMLIIARELKGGPPYCECEECKSEPPKPIVRVTLNEWADFRCGDPWPTVGTPVRALGRSLDTLPGENPDQYVALWYQSGEPVMGRIWNDGGKIAACFGWGGHEYRQKIGSIQILYELPEAIRGFDYDWKPFPEAAQFGAKEWIPVHVDHHKGNISPAVLIVDGKEILGKADIRNERATIGYGGTEKVLVGPAVHSCMVLCRKAKPGCTID
5OOA , Knot 62 124 0.80 36 93 119 
GSHMTKYKYTVEESERFNKHGIDLTVYGQVDPSATVVRVSVERGHFQEFFNVRSSYTYYVVSGQGVFYLNSEAVPAGATDLITVPPNTRIHFFGSMEMVLTVAPAFNEQDERHVRFISESESPY

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 \(c\).

\(|P_{f(4LAQ_1)}(2) \setminus P_{f(2BJR_1)}(2)|=54\), \(|P_{f(2BJR_1)}(2) \setminus P_{f(4LAQ_1)}(2)|=127\). 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 1:010100011011010001010001010010010101100111001001101000100000101000101000000001010100100000100000010000011001101001010111101111011110011100011110101100101000100010000101000011001011100000110111101010101000010100101001100100010011101101001010011000000
Pair \(Z_2\) Length of longest common subsequence
4LAQ_1,2BJR_1 181 4
4LAQ_1,5OOA_1 161 3
2BJR_1,5OOA_1 212 3

Newick tree

 
[
	2BJR_1:10.87,
	[
		4LAQ_1:80.5,5OOA_1:80.5
	]:23.37
]

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 \((0.85)(0.8)(\frac{617 }{\log_{20} 617}-\frac{249}{\log_{20}249})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 4LAQ_1 2BJR_1 130 105.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]