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Parikh vectors
4USU_1 2MAU_1 6YEI_1 Letter Amino acid
35 2 32 I Isoleucine
21 2 19 P Proline
19 1 23 T Threonine
40 2 27 V Valine
30 3 44 G Glycine
27 1 28 D Aspartic acid
31 1 23 E Glutamic acid
20 1 10 H Histidine
28 1 31 K Lycine
21 0 10 M Methionine
31 0 15 F Phenylalanine
16 3 7 Y Tyrosine
26 2 37 A Alanine
16 0 18 N Asparagine
11 6 5 C Cysteine
21 3 9 Q Glutamine
40 1 33 L Leucine
23 1 20 S Serine
5 0 6 W Tryptophan
14 0 17 R Arginine 

4USU_1|Chain A|ADENYLATE CYCLASE TYPE 10|HOMO SAPIENS (9606)
>2MAU_1|Chain A|Wrightide R1|Wrightia religiosa (429305)
>6YEI_1|Chains A, B, C, D, E, F|Glutamate dehydrogenase 1|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 \(w=f(c)\)
4USU , Knot 199 475 0.86 40 259 449 
MNTPKEEFQDWPIVRIAAHLPDLIVYGHFSPERPFMDYFDGVLMFVDISGFTAMTEKFSSAMYMDRGAEQLVEILNYHISAIVEKVLIFGGDILKFAGDALLALWRVERKQLKNIITVVIKCSLEIHGLFETQEWEEGLDIRVKIGLAAGHISMLVFGDETHSHFLVIGQAVDDVRLAQNMAQMNDVILSPNCWQLCDRSMIEIESVPDQRAVKVNFLKPPPNFNFDEFFTKCTTFMHYYPSGEHKNLLRLACTLKPDPELEMSLQKYVMESILKQIDNKQLQGYLSELRPVTIVFVNLMFEDQDKAEEIGPAIQDAYMHITSVLKIFQGQINKVFMFDKGCSFLCVFGFPGEKVPDELTHALECAMDIFDFCSQVHKIQTVSIGVASGIVFCGIVGHTVRHEYTVIGQKVNLAARMMMYYPGIVTCDSVTYNGSNLPAYFFKELPKKVMKGVADSGPLYQYWGRTEKVHHHHHH
2MAU , Knot 20 30 0.75 30 28 28 
CAQKGEYCSVYLQCCDPYHCTQPVIGGICA
6YEI , Knot 168 414 0.81 40 220 392 
SNAMNALAATNRNFKLAARLLGLDSKLEKSLLIPFREIKVECTIPKDDGTLASFVGFRVQHDNARGPMKGGIRYHPEVDPDEVNALAQLMTWKTAVAKIPYGGAKGGIGCDPSKLSISELERLTRVFTQKIHDLIGIHTDVPAPDMGTGPQTMAWILDEYSKFHGYSPAVVTGKPIDLGGSLGRDAATGRGVMFGTEALLNEHGKTISGQRFVIQGFGNVGSWAAKLISEKGGKIVAVSDITGAIKNKDGIDIPALLKHTKEHRGVKGFDGADPIDPNSILVEDCDILVPAALGGVINRENANEIKAKFIIEAANHPTDPDADEILSKKGVVILPDIYANSGGVTVSYFEWVQNIQGFMWEEEKVNDELKTYMTRSFKDLKEMCKTHSCDLRMGAFTLGVNRVAQATILRGWGA

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 \(c\).

\(|P_{f(4USU_1)}(2) \setminus P_{f(2MAU_1)}(2)|=238\), \(|P_{f(2MAU_1)}(2) \setminus P_{f(4USU_1)}(2)|=7\). 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 1:1001000100111101110110111010101001110010111111010110110001001101001100110110001011100111111011011101111110100001001101110001010111000010011010101111110101111100000011111011001011001101001110100101000011010011000110101101110101001100000110001010000110110010101010101000110011001000010101001011011110111000001001111100101010011011010100111100100110111111001100100110011011010001001001011110111101111001000001110010111011100111100001000100111011001100110111001110001100001000000
Pair \(Z_2\) Length of longest common subsequence
4USU_1,2MAU_1 245 3
4USU_1,6YEI_1 173 4
2MAU_1,6YEI_1 220 3

Newick tree

 
[
	2MAU_1:12.80,
	[
		4USU_1:86.5,6YEI_1:86.5
	]:38.30
]

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 \((0.85)(0.8)(\frac{505 }{\log_{20} 505}-\frac{30}{\log_{20}30})=147.\)
Status Protein1 Protein2 d d1/2
Query variables 4USU_1 2MAU_1 186 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]