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Parikh vectors
3SNM_1 5HYE_1 4XKM_1 Letter Amino acid
19 15 24 T Threonine
4 5 7 W Tryptophan
14 11 21 N Asparagine
18 11 33 D Aspartic acid
6 15 34 E Glutamic acid
12 20 36 K Lycine
32 19 15 S Serine
1 2 19 M Methionine
17 7 50 A Alanine
6 6 16 R Arginine
5 9 16 Q Glutamine
15 4 22 I Isoleucine
18 18 29 L Leucine
18 10 45 G Glycine
6 7 17 H Histidine
11 7 23 F Phenylalanine
7 9 18 Y Tyrosine
18 23 19 V Valine
0 7 3 C Cysteine
10 22 11 P Proline 

3SNM_1|Chain A|Concanavalin-A|Canavalia lineata (28957)
>5HYE_1|Chains A, B[auth C]|Ig gamma-1 chain C region|Homo sapiens (9606)
>4XKM_1|Chains A, B, C, D, E, F, G, H|Xylose isomerase|Bacteroides thetaiotaomicron (strain ATCC 29148 / DSM 2079 / NCTC 10582 / E50 / VPI-5482) (226186)
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 \(w=f(c)\)
3SNM , Knot 103 237 0.79 38 150 224 
ADTIVAVELDTYPNTDIGDPSYPHIGIDIKSVRSKKTAKWNMQNGKVGTAHIIYNSVGKRLSAVVSYPNGDSATVSYDVDLDNVLPEWVRVGLSASTGLYKETNTILSWSFTSKLKSNSTHETNALHFVFNQFSKDQKDLILQGDATTGTDGNLELTRVSSNGSPQGSSVGRALFYAPVHIWESSAVVASFDATFTFLIKSSDSHPADGIAFFISNIDSSIPSGSTGRLLGLFPDAN
5HYE , Knot 105 227 0.83 40 157 218 
DKTHTCPPCPAPELLGGPSVFLFPPKPKDTLMISRTPEVTCVVVDVSHEDPEVKFNWYVDGVEVHNAKTKPREEQYNSTYRVVSVLTVLHQDWLNGKEYKCKVSNKALPAPIEKTISKAKGQPREPQVYTLPPCRDELTKNQVSLWCLVKGFYPSDIAVEWESNGQPENNYKTTPPVLDSDGSFFLYSKLTVDKSRWQQGNVFSCSVMHEALHNHYTQKSLSLSPGK
4XKM , Knot 188 458 0.83 40 232 431 
MGSSHHHHHHSSGLVPRGSHMATKEFFPGIEKIKFEGKDSKNPMAFRYYDAEKVINGKKMKDWLRFAMAWWHTLCAEGGDQFGGGTKQFPWNGNADAIQAAKDKMDAGFEFMQKMGIEYYCFHDVDLVSEGASVEEYEANLKEIVAYAKQKQAETGIKLLWGTANVFGHARYMNGAATNPDFDVVARAAVQIKNAIDATIELGGENYVFWGGREGYMSLLNTDQKREKEHLAQMLTIARDYARARGFKGTFLIEPKPMEPTKHQYDVDTETVIGFLKAHGLDKDFKVNIEVNHATLAGHTFEHELAVAVDNGMLGSIDANRGDYQNGWDTDQFPIDNYELTQAMMQIIRNGGLGTGGTNFDAKTRRNSTDLEDIFIAHIAGMDAMARALESAAALLDESPYKKMLADRYASFDGGKGKEFEDGKLTLEDVVAYAKTKGEPKQTSGKQELYEAILNMYC

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 \(c\).

\(|P_{f(3SNM_1)}(2) \setminus P_{f(5HYE_1)}(2)|=73\), \(|P_{f(5HYE_1)}(2) \setminus P_{f(3SNM_1)}(2)|=80\). 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 1:100111101000100011010010111010010000010101001011010110001100101110010100101000101001110110111010011000000110101000100000000011011100100000011101010010010101001000101010011011101110110001111010101011100000011011111100100011010010111111010
Pair \(Z_2\) Length of longest common subsequence
3SNM_1,5HYE_1 153 3
3SNM_1,4XKM_1 180 4
5HYE_1,4XKM_1 217 4

Newick tree

 
[
	4XKM_1:10.28,
	[
		3SNM_1:76.5,5HYE_1:76.5
	]:29.78
]

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 \((0.85)(0.8)(\frac{464 }{\log_{20} 464}-\frac{227}{\log_{20}227})=68.7\)
Status Protein1 Protein2 d d1/2
Query variables 3SNM_1 5HYE_1 85 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]