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Parikh vectors
5VND_1 8UFE_1 7BVZ_1 Letter Amino acid
18 19 10 P Proline
20 71 27 A Alanine
17 20 14 R Arginine
20 9 19 D Aspartic acid
7 21 10 F Phenylalanine
24 10 25 E Glutamic acid
8 2 11 H Histidine
11 35 16 T Threonine
5 7 1 W Tryptophan
22 47 22 V Valine
12 11 21 N Asparagine
13 37 39 I Isoleucine
22 5 28 K Lycine
12 12 12 M Methionine
17 19 21 S Serine
12 9 10 Y Tyrosine
6 2 3 C Cysteine
9 13 7 Q Glutamine
20 57 31 G Glycine
34 64 33 L Leucine 

5VND_1|Chains A, B|Fibroblast growth factor receptor 1|Homo sapiens (9606)
>8UFE_1|Chain A|Integral membrane efflux protein EfpA|Mycolicibacterium smegmatis (1772)
>7BVZ_1|Chain A|Cell shape determining protein MreB|Spiroplasma citri (2133)
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 \(w=f(c)\)
5VND , Knot 135 309 0.83 40 198 301 
GAGVSEYELPEDPRWELPRDRLVLGKPLGEGAFGQVVLAEAIGLDKDKPNRVTKVAVKMLKSDATEKDLSDLISEMEMMKMIGKHKNIINLLGACTQDGPLYVIVECASKGNLREYLQARRPPGLEYSYNPSHNPEEQLSSKDLVSCAYQVARGMEYLASKKCIHRDLAARNVLVTEDNVMKIADFGLARDIHHIDYYKKTTNGRLPVKWMAPEALFDRIYTHQSDVWSFGVLLWEIFTLGGSPYPGVPVEELFKLLKEGHRMDKPSNCTNELYMMMRDCWHAVPSQRPTFKQLVEDLDRIVALTSNQE
8UFE , Knot 185 470 0.80 40 187 409 
WGFLSAVIAIGGMQLLATMDSTVAIVALPKIQDELSLSDAGRSWVITAYVLTFGGLMLLGGRLGDTIGRKRTFIVGVMLFTIASVLCGIAWNETTLVTARLLQGVGAAIASPTGLALVATTFPKGPARNAATAVFGAMTAIGSVMGLVVGGALTEVSWRWAFLVNVPIGLVMVYLARTALQETNRERMKLDAAGALLATLACTAAVFAFTQGPESGWLAPITLASGAAALVFGLAFLIAERNAENPVVPFALFRERNRVATFAAIFLAGGVLFTLTVLIGLYVQDILGYSALRAGVGFIPFVIGMGIGLGAASQLVRSIPPRVLVIAGGILVLGAMIYGSTLHRGIPYFPNLVLPITIGGIGIGTIVVPLTLSAIAGVNLDRIGPASAIALMLQNLGGPLVLAVIQAVITSRTLFLGGTTGPVKAMNDEQIGALDAAYTYGLLWVAAVAVLVGAAALFIGYTSQQVAHAQ
7BVZ , Knot 150 360 0.81 40 187 336 
MRPETRPFISLDLGTANVLAYVSGQGVVYNEPSLMAYNNKTNSLIALGKAAYDMVGKTHGDIRMVTPLVDGVIADMEAAQDLLKHIFSRMKMMNIWKNAIVLLACPSGVTELEREALKNVAKEMGAELVIIEEEAKMAALGAGINIELPQGHLIIDIGGGTTDLAIISSGDIVVSRSIKVAGNHFDDDIRKYIRSEYNIAIGQKTAEDVKKFIGSLVKYHNERSMQIYGRDIVSGLPKEAKISSEEIRNVLLNAFSKITDLVIELLENTPPELAGDIMRNGITVCGGGALIRNIDKYFFDIFQLPTKIASDSLNCVIEGTKIFEKTIKKNIENGLYNFQEKGLLSTLGKKRKGSHHHHHH

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 \(c\).

\(|P_{f(5VND_1)}(2) \setminus P_{f(8UFE_1)}(2)|=92\), \(|P_{f(8UFE_1)}(2) \setminus P_{f(5VND_1)}(2)|=81\). 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 1:111100001100101011000111101110111101111011110000100100111011000100001001100101101110000110111100001110111001001010001010011110000010001000100001100100110110011000010001110011100001101101111001001000000001011101111011100100000011011111101101110101111100110110010010010000001011100010111000101001100100111100000
Pair \(Z_2\) Length of longest common subsequence
5VND_1,8UFE_1 173 4
5VND_1,7BVZ_1 165 4
8UFE_1,7BVZ_1 182 4

Newick tree

 
[
	8UFE_1:90.77,
	[
		5VND_1:82.5,7BVZ_1:82.5
	]:8.27
]

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 \((0.85)(0.8)(\frac{779 }{\log_{20} 779}-\frac{309}{\log_{20}309})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 5VND_1 8UFE_1 156 133
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]