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Parikh vectors
7VWH_1 3LVV_1 9GZR_1 Letter Amino acid
2 11 1 W Tryptophan
8 30 5 Y Tyrosine
23 44 38 A Alanine
11 34 26 R Arginine
34 63 33 L Leucine
8 15 5 M Methionine
15 30 10 T Threonine
12 36 23 P Proline
15 48 20 S Serine
3 9 0 C Cysteine
13 18 7 Q Glutamine
17 43 19 E Glutamic acid
10 30 22 G Glycine
21 50 6 K Lycine
17 39 2 F Phenylalanine
17 36 29 V Valine
10 41 3 N Asparagine
13 54 17 D Aspartic acid
10 25 8 H Histidine
17 36 6 I Isoleucine 

7VWH_1|Chains A, B|Peroxisome proliferator-activated receptor delta|Homo sapiens (9606)
>3LVV_1|Chain A|Glutamate--cysteine ligase|Saccharomyces cerevisiae (4932)
>9GZR_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|RcpC|Pseudomonas aeruginosa PAO1 (208964)
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 \(w=f(c)\)
7VWH , Knot 129 276 0.87 40 186 267 
GSHMPQVADLKAFSKHIYNAYLKNFNMTKKKARSILTGKASHTAPFVIHDIETLWQAEKGLVWKQLVNGLPPYKEISVHVFYRCQCTTVETVRELTEFAKSIPSFSSLFLNDQVTLLKYGVHEAIFAMLASIVNKDGLLVANGSGFVTREFLRSLRKPFSDIIEPKFEFAVKFNALELDDSDLALFIAAIILCGDRPGLMNVPRVEAIQDTILRALEFHLQANHPDAQYLFPKLLQKMADLRQLVTEHAQMMQRIKKTETETSLHPLLQEIYKDMY
3LVV , Knot 272 692 0.85 40 293 647 
MGLLALGTPLQWFESRTYNEHIRDEGIEQLLYIFQAAGKRDNDPLFWGDELEYMVVDFDDKERNSMLDVCHDKILTELNMEDSSLCEANDVSFHPEYGRYMLEATPASPYLNYVGSYVEVNMQKRRAIAEYKLSEYARQDSKNNLHVGSRSVPLTLTVFPRMGCPDFINIKDPWNHKNAASRSLFLPDEVINRHVRFPNLTASIRTRRGEKVCMNVPMYKDIATPETDDSIYDRDWFLPEDKEAKLASKPGFIYMDSMGFGMGCSCLQVTFQAPNINKARYLYDALVNFAPIMLAFSAAAPAFKGWLADQDVRWNVISGAVDDRTPKERGVAPLLPKYNKNGFGGIAKDVQDKVLEIPKSRYSSVDLFLGGSKFFNRTYNDTNVPINEKVLGRLLENDKAPLDYDLAKHFAHLYIRDPVSTFEELLNQDNKTSSNHFENIQSTNWQTLRFKPPTQQATPDKKDSPGWRVEFRPFEVQLLDFENAAYSVLIYLIVDSILTFSDNINAYIHMSKVWENMKIAHHRDAILFEKFHWKKSFRNDTDVETEDYSISEIFHNPENGIFPQFVTPILCQKGFVTKDWKELKHSSKHERLYYYLKLISDRASGELPTTAKFFRNFVLQHPDYKHDSKISKSINYDLLSTCDRLTHLDDSKGELTSFLGAEIAEYVKKNKPSIESKCDKLAAALGHHHHHH
9GZR , Knot 114 280 0.76 38 141 254 
MNAHAPSVAPASATVDASLEKLERKPVLVASRDLPALAVIGRDDLSVELLRTAPVGSYDRPEALLGKRVWVAVPAGSILSAATLEPGGPLARTIRPDERAMAIAVDEVVGGGGFVLPGDYVDVMLFVRDERDGESTPLAQLVLPGVRVLTYGERIAVGSDGQDRSNQEKDPRPPRTAVLAVPEDGVARLMLASQAGSLRLAIRSKDEELYRREQEGALLQASLGAPSAALSLDQLLERRKPTAPSVRSAAPRVPVANGVTVYRGSAMTHEAPLEHHHHHH

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 \(c\).

\(|P_{f(7VWH_1)}(2) \setminus P_{f(3LVV_1)}(2)|=33\), \(|P_{f(3LVV_1)}(2) \setminus P_{f(7VWH_1)}(2)|=140\). 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 1:100110110101100010010100101000010011010100011111001001101001111001101111000101011000000010010010011001101001110001011001100111111101100011111010111000110010011001101010111010110100001111111111010011110110101100011011010101001010011101100110100110001011001000000001011100100010
Pair \(Z_2\) Length of longest common subsequence
7VWH_1,3LVV_1 173 4
7VWH_1,9GZR_1 155 4
3LVV_1,9GZR_1 194 6

Newick tree

 
[
	3LVV_1:96.22,
	[
		7VWH_1:77.5,9GZR_1:77.5
	]:18.72
]

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 \((0.85)(0.8)(\frac{968 }{\log_{20} 968}-\frac{276}{\log_{20}276})=186.\)
Status Protein1 Protein2 d d1/2
Query variables 7VWH_1 3LVV_1 238 165
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]