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Parikh vectors
4QHC_1 5TSJ_1 4LFP_1 Letter Amino acid
40 54 12 A Alanine
6 10 14 N Asparagine
3 8 6 W Tryptophan
3 3 8 C Cysteine
11 53 2 E Glutamic acid
10 35 6 I Isoleucine
6 19 3 F Phenylalanine
9 23 10 S Serine
23 26 7 D Aspartic acid
24 55 12 G Glycine
23 28 7 T Threonine
9 21 3 Y Tyrosine
18 56 6 V Valine
15 35 2 P Proline
19 38 11 R Arginine
7 17 3 Q Glutamine
4 6 1 H Histidine
27 47 8 L Leucine
6 24 6 K Lycine
3 19 2 M Methionine 

4QHC_1|Chain A|Beta-lactamase|Mycobacterium tuberculosis (83332)
>5TSJ_1|Chains A, B, C|V-type ATP synthase alpha chain|Thermus thermophilus (strain HB8 / ATCC 27634 / DSM 579) (300852)
>4LFP_1|Chain A|Lysozyme C|Gallus gallus (9031)
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 \(w=f(c)\)
4QHC , Knot 118 266 0.82 40 155 248 
ADLADRFAELERRYDARLGVYVPATGTTAAIEYRADERFAFCSTFKAPLVAAVLHQNPLTHLDKLITYTSDDIRSISPVAQQHVQTGMTIGQLCDAAIRYSDGTAANLLLADLGGPGGGTAAFTGYLRSLGDTVSRLDAEEPELNRDPPGDERDTTTPHAIALVLQQLVLGNALPPDKRALLTDWMARNTTGAKRIRAGFPADWKVIDKTGTGDYGRANDIAVVWSPTGVPYVVAVMSDRAGGGYDAEPREALLAEAATCVAGVLA
5TSJ , Knot 222 577 0.81 40 242 526 
MIQGVIQKIAGPAVIAKGMLGARMYDICKVGEEGLVGEIIRLDGDTAFVQVYEDTSGLKVGEPVVSTGLPLAVELGPGMLNGIYDGIQRPLERIREKTGIYITRGVVVHALDREKKWAWTPMVKPGDEVRGGMVLGTVPEFGFTHKILVPPDVRGRVKEVKPAGEYTVEEPVVVLEDGTELKMYHTWPVRRARPVQRKLDPNTPFLTGMRILDVLFPVAMGGTAAIPGPFGSGKTVTQQSLAKWSNADVVVYVGCGERGNEMTDVLVEFPELTDPKTGGPLMHRTVLIANTSNMPVAAREASIYVGVTIAEYFRDQGFSVALMADSTSRWAEALREISSRLEEMPAEEGYPPYLAARLAAFYERAGKVITLGGEEGAVTIVGAVSPPGGDMSEPVTQSTLRIVGAFWRLDASLAFRRHFPAINWNGSYSLFTSALDPWYRENVAEDYPELRDAISELLQREAGLQEIVQLVGPDALQDAERLVIEVGRIIREDFLQQNAYHEVDAYCSMKKAYGIMKMILAFYKEAEAAIKRGVSIDEILQLPVLERIGRARYVSEEEFPAYFEEAMKEIQGAFKAL
4LFP , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL

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 \(c\).

\(|P_{f(4QHC_1)}(2) \setminus P_{f(5TSJ_1)}(2)|=38\), \(|P_{f(5TSJ_1)}(2) \setminus P_{f(4QHC_1)}(2)|=125\). 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 1:10110011010000010111011101001110001000111000101111111100011001001100000010010111000100110110100111000010110111101111111011101010011001001010010100011100000001011111100111101111000111001110000110010111110101100010100101001111101011101111100011110010100111101100111111
Pair \(Z_2\) Length of longest common subsequence
4QHC_1,5TSJ_1 163 5
4QHC_1,4LFP_1 169 3
5TSJ_1,4LFP_1 208 3

Newick tree

 
[
	4LFP_1:98.77,
	[
		4QHC_1:81.5,5TSJ_1:81.5
	]:17.27
]

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 \((0.85)(0.8)(\frac{843 }{\log_{20} 843}-\frac{266}{\log_{20}266})=157.\)
Status Protein1 Protein2 d d1/2
Query variables 4QHC_1 5TSJ_1 194 140
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]