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Parikh vectors
5SNF_1 4JSJ_1 3FEV_1 Letter Amino acid
22 25 2 E Glutamic acid
10 14 0 H Histidine
21 23 4 I Isoleucine
36 36 2 L Leucine
51 20 3 A Alanine
7 18 2 Q Glutamine
17 23 3 P Proline
24 26 4 D Aspartic acid
8 27 5 K Lycine
18 19 3 F Phenylalanine
2 13 1 W Tryptophan
24 24 2 V Valine
29 22 4 R Arginine
52 25 4 G Glycine
14 11 1 M Methionine
34 26 3 S Serine
19 25 7 T Threonine
7 16 3 Y Tyrosine
15 18 4 N Asparagine
4 11 8 C Cysteine 

5SNF_1|Chains A, B|3-oxoacyl-[acyl-carrier-protein] synthase 2|Pseudomonas aeruginosa (287)
>4JSJ_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
>3FEV_1|Chains A, B, C|Fusion of Muscarinic toxin 1, Muscarinic m1-toxin1|Dendroaspis angusticeps (8618)
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 \(w=f(c)\)
5SNF , Knot 171 414 0.83 40 212 384 
MSRRRVVITGMGMLSPLGLDVPSSWEGILAGRSGIAPIEHMDLSAYSTRFGGSVKGFNVEEYLSAKEARKLDLFIQYGLAASFQAVRDSGLEVTDANRERIGVSMGSGIGGLTNIENNCRSLFEQGPRRISPFFVPGSIINMVSGFLSIHLGLQGPNYALTTAQTTGTHSIGMAARNIAYGEADVMVAGGSEMAACGLGLGGFGAARALSTRNDEPTRASRPWDRDRDGFVLSDGSGALVLEELEHARARGARIYAELVGFGMSGDAFHMTAPPEDGAGAARCMKNALRDAGLDPRQVDYINAHGTSTPAGDIAEIAAVKSVFGEHAHALSMSSTKSMTGHLLGAAGAVEAIFSVLALRDQVAPPTINLDNPDEGCDLDLVAHEAKPRKIDVALSNSFGFGGTNGTLVFRRFAD
4JSJ , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG
3FEV , Knot 38 65 0.81 38 61 63 
LTCVTSKSIFGITTENCPDGQNLCFKRWQYISPRMYDFTRGCAATCPKAEYRDVINCCGTDKCNK

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 \(c\).

\(|P_{f(5SNF_1)}(2) \setminus P_{f(4JSJ_1)}(2)|=55\), \(|P_{f(4JSJ_1)}(2) \setminus P_{f(5SNF_1)}(2)|=105\). 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 1:100001110111110111101100101111100111110010101000011101011010001010010010111001111010110001101001000011101101111100100000011001100101111110110110111010111011001100100010001111100110101011111100111011111111110110000001001001100000111100101111100100101011010101111110101101011100111110010011001110100100101010001110110111100111001011010000010101111111101110111100011110101001001001011100101001011100011111001011100110
Pair \(Z_2\) Length of longest common subsequence
5SNF_1,4JSJ_1 160 4
5SNF_1,3FEV_1 207 3
4JSJ_1,3FEV_1 243 3

Newick tree

 
[
	3FEV_1:12.85,
	[
		5SNF_1:80,4JSJ_1:80
	]:41.85
]

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 \((0.85)(0.8)(\frac{836 }{\log_{20} 836}-\frac{414}{\log_{20}414})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 5SNF_1 4JSJ_1 152 144.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]