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Parikh vectors
9EMP_1 7UCI_1 8AFL_1 Letter Amino acid
28 39 53 R Arginine
6 34 11 N Asparagine
50 107 52 L Leucine
19 32 3 P Proline
17 45 23 S Serine
16 31 32 D Aspartic acid
31 43 22 G Glycine
9 26 1 F Phenylalanine
13 27 17 T Threonine
7 7 0 W Tryptophan
27 42 19 V Valine
27 41 90 A Alanine
4 9 0 C Cysteine
18 37 35 Q Glutamine
11 42 21 I Isoleucine
8 25 1 Y Tyrosine
21 53 50 E Glutamic acid
10 29 8 H Histidine
10 28 13 K Lycine
2 16 6 M Methionine 

9EMP_1|Chain A|Probable N-acetyltransferase 16|Homo sapiens (9606)
>7UCI_1|Chains A, B|Polyketide synthase-related protein|Cylindrospermopsis raciborskii (77022)
>8AFL_1|Chains A, B, E, F|Crescentin|Caulobacter vibrioides (155892)
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 \(w=f(c)\)
9EMP , Knot 139 334 0.80 40 185 310 
MKLEPEAEAEPLDFVVATEREFEEVLAISGGIYGGLDYLPSRYHSWLRDPDRTVVLAKRNGGVIALESVNVIDAGETVLVEGLRVAPWERGKGVAGLLQRFCSQLVKRQHPGVKVARLTRDDQLGPRELKKYRLITKQGILLVRFNASALLAGLGARLAALRTSGTFSPLPTEAVSEAGGDVARLLLSPSVQRDVLPGGTIIQDWQPYRPSESNLRLLAAKGLEWRVDSRARPRVLTLCTRPFPIPHGGDGTWRYLNIDAFGSDGAQVQSQLLWHLQRQAPRLVGLNVMCQLFLEPQLWSQLADFCQVGLGLELVKGYTEQYLLEADIHHHHHH
7UCI , Knot 280 713 0.86 40 295 655 
SNAMLQKINRYTHGFVAVPVILACREKGVFELLADESPLSLNQMVEHLGANSGHFQVALRMLESLHWLSRNKELKYSLTAEAAIHNKISEDILQLYNLPIQSYLEGKQGNLLGRWIERSCQLWNLDNPLMADFLDGLLVIPLLLALHKHNLLADSEDKPLLSSLSSTVQEELGKLFLHLGWADLTAGRLTITELGRFMGERALNTAIVASYTPMLSRIHDVLFGNCLSVFQRDASGHERHIDRTLNVIGSGFQHQKYFADLEESILSVFNQLPLEEQPKYITDMGCGDGTLLKRVWETIQFKSARGKALEQYPLRLIGVDYNEASLKATTRTLASLPHLVLQGDIGNPEQMVRSLEAHGIHDPENILHIRSFLDHDRLFIPPQKRNELKERAHLPYQSVCVDDQGELIPPHVMVQSLVEHLERWSQVVNKHGLMILEVHCLEPRVVYQFLDKSENLHFDAFQGFSQQYLVEAEVFLMSAAQVGLFPKLELSKRYPKTFPFTRITLNYFEKRPYKISHAYLSDLPALVDLEVKCWPENLRASTHEIRRRLELNPQGNLVLIIEDQIIGAIYSQTITSTEALENVKYAQVPTLHTPQGSVIQLLALNILPEFQARGLGNELRDFMLYYCTLKGGIESVVGVTRCRNYVNYSQMPMMEYLKLHNEQRQLLDPIVGFHVSGGAEIRGIIANYRPEDTDNLGMGILIEYNLRDSALHS
8AFL , Knot 167 457 0.74 36 162 375 
MRLLSKNSRETKNGKPTVLGDEARAEAMQHQIESTQAIGQRYETIHGGLDSIGRVMEHLKAIEPLIAEIRGPVSQEFEARRAEHAELIAVRANLDQAQRQIALIQAEEREVSARLAAAETALGESDARRQTQDAALEDNALEIDRLRNALLQSDLKVSSLDASLRDATARIEHLVQDVEGLRVQAQDIDARRGDAEAALARANQDNALLGEEAATLKKRVDQAGLDLARLSRIETDLEAQLAAERARVQAVENALAAHQADSGRTIRGLESQVEANRAEISALQTRLETATGRADKLEEMNGQISARLADSSAQQKAVERRAGDLNVALERALDRIRALEEEADGLRQRHAGVDTARATAIERADQLAKSAVAQEKALKRAEERAQQLRARLDAMQEAQDQVRRDHEAKIAELQATIERLTSEAALAEGALEAARRDRSRLQMALLGASDGDVAASA

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 \(c\).

\(|P_{f(9EMP_1)}(2) \setminus P_{f(7UCI_1)}(2)|=29\), \(|P_{f(7UCI_1)}(2) \setminus P_{f(9EMP_1)}(2)|=139\). 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 1:1010101010110111100001001111011101110011000001100100011110001111110010110110011101101111001011111100100011000011101101000001110010000110001111101010111111110111100010101110011001110110111010100011111011001010010000101111011010100010101101000111110110101001010111001101000111010001101111011001110101100110100111110110100000110101000000
Pair \(Z_2\) Length of longest common subsequence
9EMP_1,7UCI_1 168 4
9EMP_1,8AFL_1 137 6
7UCI_1,8AFL_1 165 5

Newick tree

 
[
	7UCI_1:87.62,
	[
		9EMP_1:68.5,8AFL_1:68.5
	]:19.12
]

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 \((0.85)(0.8)(\frac{1047 }{\log_{20} 1047}-\frac{334}{\log_{20}334})=189.\)
Status Protein1 Protein2 d d1/2
Query variables 9EMP_1 7UCI_1 240 172
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]