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Parikh vectors
3KXX_1 6JYZ_1 6SZC_1 Letter Amino acid
34 35 4 L Leucine
22 3 7 K Lycine
17 20 5 S Serine
22 32 11 V Valine
12 16 1 N Asparagine
5 4 0 C Cysteine
14 14 6 H Histidine
20 62 3 A Alanine
16 30 1 R Arginine
20 33 7 D Aspartic acid
18 32 14 P Proline
11 23 20 T Threonine
5 16 0 W Tryptophan
13 13 2 Y Tyrosine
9 20 6 Q Glutamine
25 28 6 E Glutamic acid
20 54 15 G Glycine
13 17 3 I Isoleucine
14 5 1 M Methionine
7 19 6 F Phenylalanine 

3KXX_1|Chains A, B, C, D|Basic fibroblast growth factor receptor 1|Homo sapiens (9606)
>6JYZ_1|Chain A|Putative secreted endoglycosylceramidase|Rhodococcus hoagii (strain 103S) (685727)
>6SZC_1|Chain A|Surface-associated protein CshA|Streptococcus gordonii (strain Challis / ATCC 35105 / BCRC 15272 / CH1 / DL1 / V288) (467705)
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 \(w=f(c)\)
3KXX , Knot 140 317 0.84 40 200 306 
MGHHHHHHMAGVSEYELPEDPRWELPRDRLVLGKPLGEGAFGQVVLAEAIGLDKDKPNRVTKVAVKMLKSDATEKDLSDLISEMEMMKMIGKHKNIINLLGACTQDGPLYVIVEYASKGNLREYLQAREPPGLEYSYNPSHNPEEQLSSKDLVSCAYQVARGMEYLASKKCIHRDLAARNVLVTEDNVMKIADFGLARDIHHIDYYKKTTNGRLPVKWMAPEALFDRIYTHQSDVWSFGVLLWEIFTLGGSPYPGVPVEELFKLLKEGHRMDKPSNCTNELYMMMRDCWHAVPSQRPTFKQLVEDLDRIVALTSNQE
6JYZ , Knot 188 476 0.81 40 216 438 
MGSSHHHHHHSSGLVPRGSHMLPRLTVEDGAIKDVDGRTVLLRGANVNGLNDYASNGAGLPTVAPLDRTDFEAMAALGFDVVRLNIAWSALEPTPGAFDAAYVARIREAVQDAKDNGIYTVLDMHQDAWGPYVGTPEGQDCPPLLQRGIGWDGAPEWATLTGGWTTCNIGGQREASPAVARAFQAFYDDEQGVQGHLVQTWARLAAEFRNEPAVVGYDLLNEPNPGLRDPFAAADQIGRFYQRAIAAIRQAETGGFPHLVIFEPSALWSAFGFDALPPRHYLADPLVVFSPHLYSQSINVSSEFPSIEDGFRIAVAAADWYGAPLWTGEWGWFGDPDEQAGQVRRFVDAMNTHRIGGAWWSWTQACGDPHAVKDGNTAEPQGNLNRIDCPSGEEQGLVEGFAEQLARAYPRAAPGLTEVATEGFRGDGSGRIEAWYPGAERPQLDTVNVADVALTRVDGGWRLIGEAAGEYSVTTL
6SZC , Knot 55 118 0.74 36 78 108 
MAHHHHHHSSGLEVLFQGPTGTGATSTGPQGLPQTGTPTFQGGDPLVPIDETVEPTFEDGSKEKTIPGQGTYTIVPDGTVTFTPDKQFVGKPDPVTVKRVDKNGTPVTATYSPEFTKV

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 \(c\).

\(|P_{f(3KXX_1)}(2) \setminus P_{f(6JYZ_1)}(2)|=70\), \(|P_{f(6JYZ_1)}(2) \setminus P_{f(3KXX_1)}(2)|=86\). 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 1:11000000111100001100101011000111101110111101111011110000100100111011000100001001100101101110000110111100001110111001001010001010011110000010001000100001100100110110011000010001110011100001101101111001001000000001011101111011100100000011011111101101110101111100110110010010010000001011100010111000101001100100111100000
Pair \(Z_2\) Length of longest common subsequence
3KXX_1,6JYZ_1 156 6
3KXX_1,6SZC_1 180 6
6JYZ_1,6SZC_1 188 10

Newick tree

 
[
	6SZC_1:96.24,
	[
		3KXX_1:78,6JYZ_1:78
	]:18.24
]

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 \((0.85)(0.8)(\frac{793 }{\log_{20} 793}-\frac{317}{\log_{20}317})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 3KXX_1 6JYZ_1 161 135
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]