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Parikh vectors
3TTE_1 3DHS_1 6TKO_1 Letter Amino acid
19 0 9 E Glutamic acid
13 0 12 K Lycine
9 0 8 Q Glutamine
21 0 12 P Proline
7 0 11 W Tryptophan
20 0 25 I Isoleucine
43 0 42 L Leucine
12 0 16 F Phenylalanine
18 0 23 S Serine
7 0 9 Y Tyrosine
7 0 9 N Asparagine
26 0 12 D Aspartic acid
14 0 2 H Histidine
27 0 26 V Valine
37 87 16 G Glycine
13 0 9 M Methionine
21 0 20 T Threonine
49 72 33 A Alanine
19 0 22 R Arginine
1 60 12 C Cysteine 

3TTE_1|Chains A, B|Mandelate racemase/muconate lactonizing enzyme|Bradyrhizobium sp. ORS 278 (114615)
>3DHS_1|Chain A|RNase P RNA|null
>6TKO_1|Chain A|Beta-1 adrenergic receptor|Meleagris gallopavo (9103)
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 \(w=f(c)\)
3TTE , Knot 156 383 0.80 40 200 359 
MHHHHHHSSGVDLGTENLYFQSMMTTAAITGVTARAVITPMKRPLRNAFGVIDSGPLVLIDVTTDQGVTGHSYLFAYTRLALKPLVHLVEDIGRELAGKALVPVDLMKAMDAKFRLLGWQGLVGMAVSGLDMAFWDALGQLAGKPVVELLGGSARPIPAYDSYGVLDARDDERTLRTACDEHGFRAIKSKGGHGDLATDEAMIKGLRALLGPDIALMLDFNQSLDPAEATRRIARLADYDLTWIEEPVPQENLSGHAAVRERSEIPIQAGENWWFPRGFAEAIAAGASDFIMPDLMKVGGITGWLNVAGQADAASIPMSSHILPEASAHVLPVTPTAHFLEVLDFAGAILTEPLRVIDGKVTAKGPGLGLAWNESAVAKYQVT
3DHS , Knot 66 268 0.45 8 16 61 
GUUAAUCAUGCUCGGGUAAUCGCUGCGGCCGGUUUCGGCCGUAGAGGAAAGUCCAUGCUCGCACGGUGCUGAGAUGCCCGUAGUGUUCGUGGAAACACGAGCGAGAAACCCAAAUGAUGGUAGGGGCACCUUCCCGAAGGAAAUGAACGGAGGGAAGGACAGGCGGCGCAUGCAGCCUGUAGAUAGAUGAUUACCGCCGGAGUACGAGGCGCAAAGCCGCUUGCAGUACGAAGGUACAGAACAUGGCUUAUAGAGCAUGAUUAACGUC
6TKO , Knot 140 328 0.82 40 190 310 
GAELLSQQWEAGCSLLMALVVLLIVAGNVLVIAAIGRTQRLQTLTNLFITSLACADLVVGLLVVPFGATLVCRGTWLWGSFLCELWTSLDVLCVTASIWTLCVIAIDRYLAITSPFRYQSLMTRARAKVIICTVWAISALVSFLPIMMHWWRDEDPQALKCYQDPGCCDFVTNRAYAIASSIISFYIPLLIMIFVYLRVYREAKEQIRKIDRASKRKTSRVMAMREHKALKTLGIIMGVFTLCWLPFFLVNIVNVFNRDLVPKWLFVAFNWLGYANSAMNPIIYCRSPDFRKAFKRLLAARGRPLPETGGGDESATTASSSLAKDTSS

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 \(c\).

\(|P_{f(3TTE_1)}(2) \setminus P_{f(3DHS_1)}(2)|=195\), \(|P_{f(3DHS_1)}(2) \setminus P_{f(3TTE_1)}(2)|=11\). 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 1:10000000011011000101001100111011010111011001100111110011111101000011010001110001110111011001100111011111011011010101111011111110110111101110111011101111010111100001110100000010010000110110001101011000111011011111011111010001011010001101100010110011100010101110000011101100111101110111111001111011011110111011101011011100011101010111101010110110111111001101101010101111111100011100010
Pair \(Z_2\) Length of longest common subsequence
3TTE_1,3DHS_1 206 3
3TTE_1,6TKO_1 172 3
3DHS_1,6TKO_1 190 3

Newick tree

 
[
	3DHS_1:10.07,
	[
		3TTE_1:86,6TKO_1:86
	]:17.07
]

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 \((0.85)(0.8)(\frac{651 }{\log_{20} 651}-\frac{268}{\log_{20}268})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 3TTE_1 3DHS_1 155 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]