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Parikh vectors
6LTY_1 6QUF_1 3MAL_1 Letter Amino acid
2 10 5 N Asparagine
9 28 10 E Glutamic acid
10 10 11 H Histidine
5 16 11 T Threonine
1 9 6 Y Tyrosine
1 23 3 F Phenylalanine
1 6 5 W Tryptophan
21 47 12 A Alanine
0 1 2 C Cysteine
5 11 10 Q Glutamine
8 37 23 G Glycine
3 11 11 I Isoleucine
12 35 7 R Arginine
6 37 11 D Aspartic acid
13 39 13 L Leucine
2 11 17 K Lycine
4 8 2 M Methionine
0 19 7 P Proline
4 11 21 S Serine
10 19 12 V Valine 

6LTY_1|Chains A, B|Putative antitoxin HigA3|Mycobacterium tuberculosis H37Rv (83332)
>6QUF_1|Chain A|Xylose isomerase|Streptomyces rubiginosus (1929)
>3MAL_1|Chains A, B|Stromal cell-derived factor 2-like protein|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 \(w=f(c)\)
6LTY , Knot 55 117 0.74 36 80 107 
MTMARNWRDIRADAVAQGRVDLQRAAVAREEMRDAVLAHRLAEIRKALGHARQADVAALMGVSQARVSKLESGDLSHTELGTLQAYVAALGGHLRIVAEFGENTVELTALEHHHHHH
6QUF , Knot 159 388 0.81 40 206 357 
MNYQPTPEDRFTFGLWTVGWQGRDPFGDATRRALDPVESVRRLAELGAHGVTFHDDDLIPFGSSDSEREEHVKRFRQALDDTGMKVPMATTNLFTHPVFKDGGFTANDRDVRRYALRKTIRNIDLAVELGAETYVAWGGREGAESGGAKDVRDALDRMKEAFDLLGEYVTSQGYDIRFAIEPKPNEPRGDILLPTVGHALAFIERLERPELYGVNPEVGHEQMAGLNFPHGIAQALWAGKLFHIDLNGQNGIKYDQDLRFGAGDLRAAFWLVDLLESAGYSGPRHFDFKPPRTEDFDGVWASAAGCMRNYLILKERAAAFRADPEVQEALRASRLDELARPTAADGLQALLDDRSAFEEFDVDAAAARGMAFERLDQLAMDHLLGARG
3MAL , Knot 92 199 0.81 40 139 183 
GPLGSASAAASGKEGVEITYGSAIKLMHEKTKFRLHSHDVPYGSGSGQQSVTGFPGVVDSNSYWIVKPVPGTTEKQGDAVKSGATIRLQHMKTRKWLHSHLHASPISGNLEVSCFGDDTNSDTGDHWKLIIEGSGKTWKQDQRVRLQHIDTSGYLHSHDKKYQRIAGGQQEVCGIREKKADNIWLAAEGVYLPLNESSK

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 \(c\).

\(|P_{f(6LTY_1)}(2) \setminus P_{f(6QUF_1)}(2)|=27\), \(|P_{f(6QUF_1)}(2) \setminus P_{f(6LTY_1)}(2)|=153\). 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 1:101100100101011101010100111100010011110011010011101001011111110010100100101000011010101111110101110110001010110000000
Pair \(Z_2\) Length of longest common subsequence
6LTY_1,6QUF_1 180 4
6LTY_1,3MAL_1 143 3
6QUF_1,3MAL_1 169 4

Newick tree

 
[
	6QUF_1:91.95,
	[
		6LTY_1:71.5,3MAL_1:71.5
	]:20.45
]

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 \((0.85)(0.8)(\frac{505 }{\log_{20} 505}-\frac{117}{\log_{20}117})=115.\)
Status Protein1 Protein2 d d1/2
Query variables 6LTY_1 6QUF_1 142 89.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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