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Parikh vectors
7YAZ_1 3WTX_1 3PAR_1 Letter Amino acid
12 16 10 T Threonine
9 5 10 Y Tyrosine
19 15 8 A Alanine
11 11 8 N Asparagine
15 10 8 D Aspartic acid
10 6 2 H Histidine
22 13 9 V Valine
13 18 5 R Arginine
7 2 4 C Cysteine
22 8 7 K Lycine
7 2 2 W Tryptophan
27 17 8 S Serine
9 13 8 Q Glutamine
17 10 14 G Glycine
19 5 7 I Isoleucine
10 4 5 M Methionine
27 7 13 E Glutamic acid
29 15 11 L Leucine
14 8 4 F Phenylalanine
11 19 5 P Proline 

7YAZ_1|Chain A|Mitogen-activated protein kinase kinase kinase MLT|Homo sapiens (9606)
>3WTX_1|Chains A, D[auth F]|Runt-related transcription factor 1|Mus musculus (10090)
>3PAR_1|Chain A|Pulmonary surfactant-associated protein A|Rattus norvegicus (10116)
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 \(w=f(c)\)
7YAZ , Knot 136 310 0.84 40 202 298 
GAMGSGASFVQIKFDDLQFFENCGGGSFGSVYRAKWISQDKEVAVKKLLKIEKEAEILSVLSHRNIIQFYGVILEPPNYGIVTEYASLGSLYDYINSNRSEEMDMDHIMTWATDVAKGMHYLHMEAPVKVIHRDLKSRNVVIAADGVLKICDFGASRFHNHTTHMSLVGTFPWMAPEVIQSLPVSETCDTYSYGVVLWEMLTREVPFKGLEGLQVAWLVVEKNERLTIPSSCPRSFAELLHQCWEADAKKRPSFKQIISILESMSNDTSLPDKCNSFLHNKAEWRCEIEATLERLKKLERDLSFKEQELK
3WTX , Knot 100 204 0.87 40 152 200 
GELVRTDSPNFLCSVLPTHWRCNKTLPIAFKVVAKGDVPDGTLVTVMAGNDENYSAELRNATAAMKNQVARFNDLRFVGRSGRGKSFTLTITVFTNPPQVATYHRAIKITVDGPREPRRHRQKLDDQTKPGSLSFSERLSELEQLRRTAMRVSPHHPAPTPNPRASLNHSTAFNPQPQSQMQDARQIQPSPPWSYDQSYQYLGS
3PAR , Knot 71 148 0.80 40 117 144 
AYLDEELQTELYEIKHQILQTMGVLSLQGSMLSVGDKVFSTNGQSVNFDTIKEMCTRAGGNIAVPRTPEENEAIASIAKKYNNYVYLGMIEDQTPGDFHYLDGASVSYTNWYPGEPRGQGKEKCVEMYTDGTWNDRGCLQYRLAVCEF

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 \(c\).

\(|P_{f(7YAZ_1)}(2) \setminus P_{f(3WTX_1)}(2)|=118\), \(|P_{f(3WTX_1)}(2) \setminus P_{f(7YAZ_1)}(2)|=68\). 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 1:1111011011010100101100011101101001011000001110011010001011011000011010111101100111000101101000100000001010011011001101100101011101100010000111110111010011100100000010111011111101100111000000000111110110001110110110111111000001011000100110110001010100010100110110010000011000001100010100010101001001000101000010
Pair \(Z_2\) Length of longest common subsequence
7YAZ_1,3WTX_1 186 4
7YAZ_1,3PAR_1 183 3
3WTX_1,3PAR_1 169 3

Newick tree

 
[
	7YAZ_1:94.69,
	[
		3PAR_1:84.5,3WTX_1:84.5
	]:10.19
]

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 \((0.85)(0.8)(\frac{514 }{\log_{20} 514}-\frac{204}{\log_{20}204})=89.5\)
Status Protein1 Protein2 d d1/2
Query variables 7YAZ_1 3WTX_1 116 95.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]