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Parikh vectors
1UUM_1 1XDV_1 1AMF_1 Letter Amino acid
10 41 7 N Asparagine
38 33 16 G Glycine
13 23 3 H Histidine
3 20 2 M Methionine
22 32 14 T Threonine
8 30 7 Y Tyrosine
21 34 16 D Aspartic acid
0 14 0 C Cysteine
18 37 8 Q Glutamine
0 7 3 W Tryptophan
32 43 23 V Valine
28 51 6 R Arginine
14 63 10 I Isoleucine
47 84 17 L Leucine
19 52 15 S Serine
33 41 34 A Alanine
20 92 12 E Glutamic acid
17 63 21 K Lycine
10 43 9 F Phenylalanine
19 44 10 P Proline 

1UUM_1|Chains A, B|DIHYDROOROTATE DEHYDROGENASE|RATTUS RATTUS (10117)
>1XDV_1|Chains A, B|Son of sevenless protein homolog 1|Homo sapiens (9606)
>1AMF_1|Chain A|MOLYBDATE TRANSPORT PROTEIN MODA|Escherichia coli (562)
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 \(w=f(c)\)
1UUM , Knot 155 372 0.82 36 198 348 
HHHHHHATGDDHFYAEYLMPGLQRLLDPESAHRLAVRVTSLGLLPRATFQDSDMLEVKVLGHKFRNPVGIAAGFDKNGEAVDGLYKLGFGFVEVGSVTPQPQEGNPRPRVFRLPEDQAVINRYGFNSHGLSVVEHRLRARQQKQAQLTADGLPLGINLGKNKTSEDAAADYAEGVRTLGPLADYLVVNVSSPNTAGLRSLQGKTELRHLLSKVLQERDALKGTRKPAVLVKIAPDLTAQDKEDIASVARELGIDGLIVTNTTVSRPVGLQGALRSETGGLSGKPLRDLSTQTIREMYALTQGRIPIIGVGGVSSGQDALEKIQAGASLVQLYTALIFLGPPVVVRVKRELEALLKERGFTTVTDAIGADHRR
1XDV , Knot 322 847 0.85 40 316 770 
EQTYYDLVKAFMAEIRQYIRELNLIIKVFREPFVSNSKLFSANDVENIFSRIVDIHELSVKLLGHIEDTVEMTDEGSPHPLVGSCFEDLAEELAFDPYESYARDILRPGFHDRFLSQLSKPGAALYLQSIGEGFKEAVQYVLPRLLLAPVYHCLHYFELLKQLEEKSEDQEDKECLKQAITALLNVQSGMEKICSKSLAKRRLSESACRFYSQQMKGKQLAIKKMNEIQKNIDGWEGKDIGQCCNEFIMEGTLTRVGAKHERHIFLFDGLMICCKSNHGQPRLPGASNAEYRLKEKFFMRKVQINDKDDTNEYKHAFEIILKDENSVIFSAKSAEEKNNWMAALISLQYRSTLERMLDVTMLQEEKEEQMRLPSADVYRFAEPDSEENIIFEENMQPKAGIPIIKAGTVIKLIERLTYHMYADPNFVRTFLTTYRSFCKPQELLSLIIERFEIPEPEPTEADRIAIENGDQPLSAELKRFRKEYIQPVQLRVLNVCRHWVEHHFYDFERDAYLLQRMEEFIGTVRGKAMKKWVESITKIIQRKKIARDNGPGHNITFQSSPPTVEWHISRPGHIETFDLLTLHPIEIARQLTLLESDLYRAVQPSELVGSVWTKEDKEINSPNLLKMIRHTTNLTLWFEKCIVETENLEERVAVVSRIIEILQVFQELNNFNGVLEVVSAMNSSPVYRLDHTFEQIPSRQKKILEEAHELSEDHYKKYLAKLRSINPPCVPFFGIYLTNILKTEEGNPEVLKRHGKELINFSKRRKVAEITGEIQQYQNQPYCLRVESDIKRFFENLNPMGNSMEKEFTDYLFNKSLEIEPRNPKPLPRFPKKYSYPLKSPGVRPSN
1AMF , Knot 104 233 0.81 38 150 227 
DEGKITVFAAASLTNAMQDIATQFKKEKGVDVVSSFASSSTLARQIEAGAPADLFISADQKWMDYAVDKKAIDTATRQTLLGNSLVVVAPKASVQKDFTIDSKTNWTSLLNGGRLAVGDPEHVPAGIYAKEALQKLGAWDTLSPKLAPAEDVRGALALVERNEAPLGIVYGSDAVASKGVKVVATFPEDSHKKVEYPVAVVEGHNNATVKAFYDYLKGPQAAEIFKRYGFTIK

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 \(c\).

\(|P_{f(1UUM_1)}(2) \setminus P_{f(1XDV_1)}(2)|=23\), \(|P_{f(1XDV_1)}(2) \setminus P_{f(1UUM_1)}(2)|=141\). 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 1:000000101000101001111100110100100111010011111010100001101011100100111111110001011011001111110110101010010101011011000111000110001101100010100000101010111111011000000011100101100111110011101001001110010100010011001100001101000111110111010100000110110011101111000010011110111000011101011001000010010110010111111111001001100101110110100111111111110100010111000110010011110000
Pair \(Z_2\) Length of longest common subsequence
1UUM_1,1XDV_1 164 4
1UUM_1,1AMF_1 134 4
1XDV_1,1AMF_1 200 4

Newick tree

 
[
	1XDV_1:98.24,
	[
		1UUM_1:67,1AMF_1:67
	]:31.24
]

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 \((0.85)(0.8)(\frac{1219 }{\log_{20} 1219}-\frac{372}{\log_{20}372})=221.\)
Status Protein1 Protein2 d d1/2
Query variables 1UUM_1 1XDV_1 284 200.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]