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Parikh vectors
5KJK_1 5AKV_1 4PMH_1 Letter Amino acid
41 12 8 E Glutamic acid
22 10 33 G Glycine
16 4 3 H Histidine
35 8 14 K Lycine
13 1 3 M Methionine
20 4 10 R Arginine
17 1 3 C Cysteine
15 0 13 Q Glutamine
17 5 10 F Phenylalanine
4 2 6 W Tryptophan
41 7 15 L Leucine
20 5 24 Y Tyrosine
28 12 34 A Alanine
15 3 28 N Asparagine
21 5 18 D Aspartic acid
13 12 38 T Threonine
25 12 19 V Valine
20 5 23 I Isoleucine
19 8 16 P Proline
27 11 48 S Serine 

5KJK_1|Chain A|N-lysine methyltransferase SMYD2|Homo sapiens (9606)
>5AKV_1|Chains A, B|TRANSTHYRETIN|Homo sapiens (9606)
>4PMH_1|Chain A|Pectinesterase|Sitophilus oryzae (7048)
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 \(w=f(c)\)
5KJK , Knot 179 429 0.84 40 239 406 
GLGGLERFCSPGKGRGLRALQPFQVGDLLFSCPAYAYVLTVNERGNHCEYCFTRKEGLSKCGRCKQAFYCNVECQKEDWPMHKLECSPMVVFGENWNPSETVRLTARILAKQKIHPERTPSEKLLAVKEFESHLDKLDNEKKDLIQSDIAALHHFYSKHLEFPDNDSLVVLFAQVNCNGFTIEDEELSHLGSAIFPDVALMNHSCCPNVIVTYKGTLAEVRAVQEIKPGEEVFTSYIDLLYPTEDRNDRLRDSYFFTCECQECTTKDKDKAKVEIRKLSDPPKAEAIRDMVRYARNVIEEFRRAKHYKSPSELLEICELSQEKMSSVFEDSNVYMLHMMYQAMGVCLYMQDWEGALQYGQKIIKPYSKHYPLYSLNVASMWLKLGRLYMGLEHKAAGEKALKKAIAIMEVAHGKDHPYISEIKQEIESH
5AKV , Knot 65 127 0.82 38 101 125 
GPTGTGESKCPLMVKVLDAVRGSPAINVAVHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGISPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNPKE
4PMH , Knot 150 366 0.80 40 191 339 
DQTAPGTASRPILTASESNYFTTATYLQGWSPPSISTSKADYTVGNGYNTIQAAVNAAINAGGTTRKYIKINAGTYQEVVYIPNTKVPLTIYGGGSSPSDTLITLNMPAQTTPSAYKSLVGSLFNSADPAYSMYNSCASKSGTIGTSCSTVFWVKAPAVQIVNLSIENSAKNTGDQQAVALQTNSDQIQIHNARLLGHQDTLYAGSGSSSVERSYYTNTYIEGDIDFVFGGGSAIFESCTFYVKADRRSDTAVVFAPDTDPHKMYGYFVYKSTITGDSAWSSSKKAYLGRAWDSGVSSSSAYVPGTSPNGQLIIKESTIDGIINTSGPWTTATSGRTYSGNNANSRDLNNDNYNRFWEYNNSGNGA

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 \(c\).

\(|P_{f(5KJK_1)}(2) \setminus P_{f(5AKV_1)}(2)|=164\), \(|P_{f(5AKV_1)}(2) \setminus P_{f(5KJK_1)}(2)|=26\). 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 1:111110010011010110110110110111001101011010001000000100001100010000110001000000111001000111111001010001010101110001010001000111100100010010000001100011110010000101100001111110100011010000100110111101111000001011100010110101100101100110001011010000000100001100000000000000101010010011010110011001001100100100000100110100100001001100001011011001111010100101110010011010000011001011011101101011100011100110011111011010001010010001000
Pair \(Z_2\) Length of longest common subsequence
5KJK_1,5AKV_1 190 4
5KJK_1,4PMH_1 178 5
5AKV_1,4PMH_1 176 4

Newick tree

 
[
	5KJK_1:93.35,
	[
		4PMH_1:88,5AKV_1:88
	]:5.35
]

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 \((0.85)(0.8)(\frac{556 }{\log_{20} 556}-\frac{127}{\log_{20}127})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 5KJK_1 5AKV_1 161 102
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]