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Parikh vectors
6NRH_1 3PXV_1 5IDI_1 Letter Amino acid
25 16 29 L Leucine
24 8 27 K Lycine
8 6 24 F Phenylalanine
13 10 19 P Proline
6 7 10 Q Glutamine
12 15 33 E Glutamic acid
19 12 34 I Isoleucine
24 9 18 S Serine
1 1 15 W Tryptophan
12 21 26 A Alanine
12 7 27 N Asparagine
15 2 20 H Histidine
26 17 37 G Glycine
6 10 7 M Methionine
14 8 15 T Threonine
12 8 27 Y Tyrosine
16 10 33 V Valine
9 9 22 R Arginine
15 7 28 D Aspartic acid
2 6 1 C Cysteine 

6NRH_1|Chain A|Poly [ADP-ribose] polymerase 1|Homo sapiens (9606)
>3PXV_1|Chains A, B, C, D|Nitroreductase|Desulfitobacterium hafniense (272564)
>5IDI_1|Chains A, B|1,4-beta-D-glucan glucohydrolase|Thermotoga neapolitana (2337)
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 \(w=f(c)\)
6NRH , Knot 120 271 0.82 40 172 254 
MGSSHHHHHHSSGLVPRGSHMTKSKLPKPVQDLIKMIFGSGSGSGGDPIDVNYEKLKTDIKVVDRDSEEAEIIRKYVKNTHATTHNAYDLEVIDIFKIEREGECQRYKPFKQLHNRRLLWHGSRTTNFAGILSQGLRIAPPEAPVTGYMFGKGIYFADMVSKSANYCHTSQGDPIGLILLGEVALGNMYELKHASHISKLPKGKHSVKGLGKTTPDPSANISLDGVDVPLGTGISSGVNDTSLLYNEYIVYDIAQVNLKYLLKLKFNFKTS
3PXV , Knot 92 189 0.85 40 140 184 
GMMNETLKVIAERYSCRDFKNEMPSDELLQAIAEAAIQAPSGMNRQAWRVIVVKNKELMQEMEAEGLAYLAGMEDQSSYNRIMERGGRLFYGAPCMIVVPIDPTQYGPALVDCGILCQTIALAATSLGIANIMCGYTGLAFASGLRAEEFSKRLGFPEGYAFGCSVLLGHANTTKPPHVPDKDKITYVE
5IDI , Knot 187 452 0.84 40 245 431 
MKKFPEGFLWGVATASYQIEGSPLADGAGMSIWHTFSHTPGNVKNGDTGDVACDHYNRWKEDIEIIEKIGAKAYRFSISWPRILPEGTGKVNQKGLDFYNRIIDTLLEKNITPFITIYHWDLPFSLQLKGGWANRDIADWFAEYSRVLFENFGDRVKHWITLNELWVVAIVGHLYGVHAPGMKDIYVAFHTVHNLLRAHAKSVKVFRETVKDGKIGIVFNNGYFEPASEREEDIRAARFMHQFNNYPLFLNPIYRGEYPDLVLEFAREYLPRNYEDDMEEIKQEIDFVGLNYYSGHMVKYDPNSPARVSFVERNLPKTAMGWEIVPEGIYWILKGVKEEYNPQEVYITGNGAAFDDVVSEGGKVHDQNRIDYLRAHIEQVWRAIQDGVPLKGYFVWSLLDNFEWAEGYSKRFGIVYVDYNTQKRIIKDSGYWYSNVIKNNGLTDLEHHHHHH

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 \(c\).

\(|P_{f(6NRH_1)}(2) \setminus P_{f(3PXV_1)}(2)|=104\), \(|P_{f(3PXV_1)}(2) \setminus P_{f(6NRH_1)}(2)|=72\). 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 1:1100000000001111010010000110110011011110101011011010000100010110000001011000100001000010010110110100010000001100100001110100000111110011011110111010111011011011000100000001011111111011110100100100100110100010111000101010101011011110110011000011000011001101010011010101000
Pair \(Z_2\) Length of longest common subsequence
6NRH_1,3PXV_1 176 3
6NRH_1,5IDI_1 167 6
3PXV_1,5IDI_1 205 4

Newick tree

 
[
	3PXV_1:99.21,
	[
		6NRH_1:83.5,5IDI_1:83.5
	]:15.71
]

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 \((0.85)(0.8)(\frac{460 }{\log_{20} 460}-\frac{189}{\log_{20}189})=79.3\)
Status Protein1 Protein2 d d1/2
Query variables 6NRH_1 3PXV_1 101 85
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]