King M.R., Mody N.A. Numerical and Statistical Methods for Bioengineering: Applications in MATLAB

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a few diff erent values of α. Choose α

max

such that zero and α

max

bracket the

minimum along the direction of the search. Perform this exercise using the

starting point (8, 2) and use your chosen value of α

max

for all iterations.

(b) Write a function m-ﬁle that performs the multidimensional minimization using

steepest ascent. You can use the steepestparabolicinterpolation func-

tion listed in Program 8.10 to perform the one-dimensional minimization.

tolerances: tolerance for α ¼ 0:0001; tolerance for krf k¼ 0:01.

8.5. Friction coefﬁcient of a n ellipsoid moving through a ﬂuid Many microorganisms,

such as bacteria, have an elongated, ellipsoidal shape. The friction coefﬁcient ξ

relates the force on a particle and its velocity when moving through a viscous

ﬂuid – thus, it is an important parameter in centrifugation assays. A lower friction

coefﬁcient indicates that a particle moves “easier” through a ﬂuid. The friction

coefﬁcient for an ellipsoid moving in the direction of its long axis, as shown in

Figure P8.2, is given by

ξ ¼

4πμa

lnð2a=bÞ1=2

;

where μ is the ﬂuid viscosity (for water, this is equal to 1 μg/(s μm)) (Saltzman, 2001).

For a short axis of b =1μm, ﬁnd the value of a that corresponds to a minimum in the

friction coefﬁcient in water. Start with an initial guess of a = 1.5 μm, and show the

ﬁrst three iterations.

(a) Use the golden section search method to search for the optimum.

(b) Check your answer by using Newton’s method to ﬁnd the optimum.

8.6. Nonlinear regression of chemical reaction data An experimenter examines the

reaction kinetics of an NaOH:phenolphthale in reaction. The extent of reaction is

measured using a colorimeter, where the absorbance is proportional to the concen-

tration of the phenolphthalein. The purpose is to measure the reaction rate constant

k at different temperatures and NaOH concentrations to learn about the reaction

mechanism. In the experiment, the absorbances A

are measured at times t

. The

absorbance should ﬁt an exponential rate law:

A ¼ C

þ C

kt

;

where C

, C

, and k are unknow n constants.

(a) Write a MATLAB function that can be fed into the built-in function

fminsearch to perform a nonlinear regression for the unknown con stants.

(b) Now suppose that you obtain a solution, but you suspect that the fminsearch

routine may be getting stuck in a local minimum that is not the most accurate

solution. How would you test this?

8.7. Nonlinear regression of microﬂuidic velocity data In the entrance region of a micro-

ﬂuidic channel, some distance is required for the ﬂow to adjust from upstream

conditions to the fully developed ﬂow pattern. This distance depends on the ﬂow

conditions upstream and on the Reynolds number (Re ¼ Dvρ=μ), which is a

Figure P8.2

537

8.7 Problems

dimensionless group that characterizes the relative importance of inertial ﬂuid forces

to viscous ﬂuid forces. For a uniform velocity proﬁle at the channel entrance, the

computed length in laminar ﬂow (entrance length relative to hydraulic diameter =

ent

) required for the centerline velocity to reach 99% of its fully developed value

is given by

ent

¼ a  exp b  ReðÞþc Re þ d:

Suppose that you have used micro-particle image velocimetry (μ-PIV) to measure

this entrance length as a function of Reynolds numb er in your microﬂuidic system,

and would like to perform a nonlinear regres sion on the data to determine the four

model parameters a, b, c,andd. You woul d like to perform nonlinear regression

using the method of steepest descent.

(a) What is the objective function to be minimized?

(b) Find the gradient vector.

b, c,andd.

8.8. Using hemoglobin as a blood substitute: hemoglobin–oxygen binding Find the

best-ﬁt values for the Hill parameters P

and n in Equation 2.27.

(a) Use the fminsearch function to ﬁnd optimal values that correspond to the

least-squared error.

(b) Compare your optimal values with those obtained us ing linear regression of the

transformed equation.

parameters. Create ten data sets from the original data set. Calculate the stand-

ard deviation for each parameter.

References

Boal, J. H., Plessinger, M. A., van den Reydt, C., and Miller, R. K. (1997) Pharmacokinetic

and Toxicity Studies of AZT (Zidovudine) Following Perfusion of Human Term

Placenta for 14 Hours. Toxicol. Appl. Pharmacol., 143, 13–21.

Edgar, T. F., and Himmelblau, D. M. (1988) Optimization of Chemical Processes (New York:

McGraw-Hill).

Fournier, R. L. (2007) Basic Transport Phenomena in Biomedical Engineering (New York:

Taylor & Francis).

Nauman, B. E. (2002) Chemical Reactor Design, Optimization, and Scaleup (New York:

McGraw-Hill).

Rao, S. S. (2002) Applied Numerical Methods for Engineers and Scientists (Upper Saddle

River, NJ: Prentice Hall).

Saltzman, W. M. (2001) Drug Delivery: Engineering Principles for Drug Therapy (New York:

Oxford University Press).

538

Nonlinear model regression and optimization

9 Basic algorithms of bioinformatics

9.1 Introduction

The primary goal of the ﬁeld of bioinformatics is to examine the biologically impor-

tant information that is stored, used, and transferred by living things, and how this

information acts to control the chemical environment within living organisms. This

work has led to technological successes such as the rapid development of potent

HIV-1 proteinase inhibitors, the development of new hybrid seeds or genetic varia-

tions for impr oved agriculture, and even to new understanding of pre-historical

patterns of human migration. Before discussing useful algorithms for bioinformatic

analysis, we must ﬁrst cover some preliminary concepts that will allow us to speak a

common language.

Deoxyribonucleic acid (DNA) is the genetic material t hat is passed down from

parent to offspring. DNA is stored in the nuclei of cells and forms a complete set of

instructions for the growth, development, and functioning of a living organism. As

a macromolecule, DNA can contain a vast amount of information through a

speciﬁc sequence of bases: guanine (G), adenine (A), thymine (T), and cytosine

(C). Each base is attached to a phosphate group and a deoxyribose sugar to form a

nucleotide unit. The four different nucleotides are then strung into long polynu-

cleotide chains, which comprise genes that are thousands of bases long, and

ultimately into chromosomes. In humans, the molecule of DNA in a single chro-

mosome ranges from 50 × 10

nucleotide pairs in the smallest chromosome, up to

250 × 10

in the largest. If stretched end-to-end, these DNA chains would extend

1.7 cm to 8.5 cm, respectively! The directional orientation of the DNA sequence

has great biological signiﬁcance i n the duplication and translation processes. DNA

sequences proceed from the 5

carbon phosphate group end t o the 3

carbon of the

deoxyribose sugar end.

Aspresentedinalandmark1953paper,WatsonandCrickdiscovered,from

X-ray crystallographic data combined with theoretical model building, that the

four bases of DNA undergo complementary hydrogen binding with t he appro-

priate base on a neighboring strand, which causes complementary sequences of

DNA to assemble into a double helical structure. T he four bases, i.e. the basic

building blocks of DNA, are classiﬁed as either purines (A and G) or pyrimidines

(T and C), which comprise either a double or single ring structure (respectively) of

carbon, nitrogen, hydrogen, and oxygen. A purine can form hydrogen bonds only

with its complementary pyrimidine base pair. Speciﬁcally, when in the proper

orientation and separated by a distance of 11 A

, guanine (G) will share three

hydrogen bonds with cytosine (C), whereas adenine (A) will share t wo hydrogen

bonds with thymine (T). This phenomenon is called complementary base

pairing. The two paired strands that form the D NA double helix run anti-parallel

to each other, i.e. the 5

end of one strand faces the 3

end of the complementary

strand. One DNA chain can serve as a template for the assembly of a second,

complementary DNA strand, which is precisely how the genome is replicated

prior to cell division. Complementary DNA will spontaneously pair up (or

“hybridize”) at room or body temperature, and can be caused to separate or

“melt” at elevated temperature, which was later exploited in the widely used

polymerase chain reaction (PCR) m ethod of DNA ampliﬁcation in the

laboratory.

So how does the genome, encoded in the sequen ce of DNA, become the thousands

of proteins within the cell that interact and participate in chemical reactions? It has

been determined that each gene corresponds in general to one protein. The process

by which the genes encoded in DNA are made into the myriad of proteins found in

the cell is referred to as the “central dogma” of molecular biology. Information

contained in DNA is “transcribed” into the single-stranded polynucleotide ribonu-

cleic acid (RNA) by RNA polymerases. The nucleotide sequences in DNA are

copied directly into RNA, except for thymine (T), which is substituted with uracil

(U, a pyrimidine) in the new RNA molec ule. The RNA gene sequences are then

“translated” into the amino acid sequences of different proteins by protein/RNA

complexes known as ribosomes. Groups of three nucleotides, each called a “codon”

(N =4

= 64 combinations), encode the 20 different amino acids that are the

building blocks of proteins. Table 9.1 shows the 64 codons and the corresponding

amino acid or start/stop signal.

The three stop codons, UAA, UAG, and UGA, which instruct ribosomes to

terminate the translation, have been given the names “ochre,” “amber,” and

“opal,” respectively. The AUG codon, encoding for methionine, represents the

translation start signal. Not all of the nucleotides in DNA encode for proteins,

however. Large stretches of the genome, called introns, are spliced out of the

sequence by enzyme complexes which recognize the proper splicing signals, and

the remaining exons are joined together to form the protein-encoding portions.

Major sequence repositories are curated by the National Center for Biotechnology

Information (NCBI) in the United States, the European Molecular Biology

Laboratory (EMBL), and the DNA Data Bank of Japan.

9.2 Sequence alignment and database searches

The simplest method for identifying regions of similarity between two sequences

is to produce a graphical dot plot. A two-dimensional graph is generated, and a

dot is placed at each position where the two compared sequences are identical.

The word size can be speciﬁed, as in MATL AB program 9.1, to reduce the

noisiness produced by many very short (length = 1–2) regions of similarity.

Identity runs along the main diagonal, and common subsequences including

possible translocations are seen as shorter off-diagonal lines. Inversions,wherea

region of the gene runs in the opposite direction, appear as lines perpendicular to

the main diagonal, while deletions appear as interruptions in the lines. Figure 9.1

shows the sequence of human P-selectin, an adhesion protein important in

inﬂammation, compared against itself. In the P-selectin secondary structure,

it is known that nine consensus repeat domains

exist, and these can be seen as

A consensus repeat domain is a sequence of amino acids that occurs with high frequency in a

polypeptide.

540

Basic algorithms of bioinformatics

Table 9.1. Genetic code for translation from mRNA to amino acids

2nd base

UCAG

1st base

U UUU: phenylalanine (Phe/F)

UUC: phenylalanine (Phe/F)

UUA: leucine (Leu/L)

UUG: leucine (Leu/L)

UCU: serine (Ser/S)

UCC: serine (Ser/S)

UCA: serine (Ser/S)

UCG: serine (Ser/S)

UAU: tyrosine (Tyr/Y)

UAC: tyrosine (Tyr/Y)

UAA: stop (“ochre”)

UAG: stop (“amber”)

UGU: cysteine (Cys/C)

UGC: cysteine (Cys/C)

UGA: stop (“opal”)

UGG: tryptophan (Trp/W)

C CUU: leucine (Leu/L)

CUC: leucine (Leu/L)

CUA: leucine (Leu/L)

CUG: leucine (Leu/L)

CCU: proline (Pro/P)

CCC: proline (Pro/P)

CCA: proline (Pro/P)

CCG: proline (Pro/P)

CAU: histidine (His/H)

CAC: histidine (His/H)

CAA: glutamine (Gln/Q)

CAG: glutamine (Gln/Q)

CGU: arginine (Arg/R)

CGC: arginine (Arg/R)

CGA: arginine (Arg/R)

CGG: arginine (Arg/R)

A AUU: isoleucine (Ile/I)

AUC: isoleucine (Ile/I)

AUA: isoleucine (Ile/I)

AUG: start, methionine (Met/M)

ACU: threonine (Thr/T)

ACC: threonine (Thr/T)

ACA: threonine (Thr/T)

ACG: threonine (Thr/T)

AAU: asparagine (Asn/N)

AAC: asparagine (Asn/N)

AAA: lysine (Lys/K)

AAG: lysine (Lys/K)

AGU: serine (Ser/S)

AGC: serine (Ser/S)

AGA: arginine (Arg/R)

AGG: arginine (Arg/R)

G GUU: valine (Val/V)

GUC: valine (Val/V)

GUA: valine (Val/V)

GUG: valine (Val/V)

GCU: alanine (Ala/A)

GCC: alanine (Ala/A)

GCA: alanine (Ala/A)

GCG: alanine (Ala/A)

GAU: aspartic acid (Asp/D)

GAC: aspartic acid (Asp/D)

GAA: glutamic acid (Glu/E)

GAG: glutamic acid (Glu/E)

GGU: glycine (Gly/G)

GGC: glycine (Gly/G)

GGA: glycine (Gly/G)

GGG: glycine (Gly/G)

off-diagonal parallel lines in Figure 9.1(b) once the spurious dots are ﬁltered out

by specifying a word size of 3. Website tools are availabl e (European Molecular

Biology Open Software Suite, EMBOSS) to make more sophisticated dot plot

comparisons.

Box 9.1 Sequence formats

When retrieving genetic sequences from online databases for analysis using commonly available web

tools, it is important to work in specific sequence formats. One of the most common genetic sequence

formats is called Fasta. Let’s find the amino acid sequence for ADAM metallopeptidase domain 17

(ADAM17), a transmembrane protein responsible for the rapid cleavage of L-selectin from the leukocyte

surface (see Problem 1.14 for a brief introduction to selectin-mediated binding of flowing cells to the

vascular endothelium), in the Fasta format. First we go to the Entrez cross-database search page at the

NCBI website (www.ncbi.nlm.nih.gov/sites/gquery), a powerful starting point for accessing the wealth

of genomic data available online. By clicking on the “Protein” link, we will restrict our search to the

protein sequence database. Entering “human ADAM17” in the search window produces 71 hits, with the

first being accession number AAI46659 for ADAM17 protein [Homo sapiens]. Clicking on the AAI46659

link takes us to a page containing information on this entry, including the journal citation in which the

sequence was first published. Going to the “Display” pulldown menu near the top of the page, we select

the “FASTA” option and the following text is displayed on the screen:

>gi|148922164|gb|AAI46659.1| ADAM17 protein [Homo sapiens]

MRQSLLFLTSVVPFVLAPRPPDDPGFGPHQRLEKLDSLLSDYDILSLSNI

QQHSVRKRDLQTSTHVETLLTFSALKRHFKLYLTSSTERFSQNF

KVVVVDGKNESEYTVKWQDFFTGHVVGEPDSRVLAHIRDDDVIIRI

NTDGAEYNIEPLWRFVNDTKDKRMLVYKSEDIKNVSRLQSPKVCGYLKVDN

EELLPKGLVDREPPEELVHRVKRRADPDPMKNTCKLLVVADHRF

YRYMGRGEESTTTNYLIHTDRAN

This is the amino acid sequence of human ADAM17, displayed in Fasta format. It is composed of a

greater than symbol, no space, then a description line, followed by a carriage return and the sequence in

single-letter code. The sequence can be cut and pasted from the browser window, or output to a text file.

We may also choose to output a specified range of amino acids rather than the entire sequence. If we

instead select the “GenPept” option from the Display pulldown menu (called “GenBank” format for DNA

sequences) then we see the original default listing for this accession number. At the bottom of the

screen, after the citation information, is the same ADAM17 sequence in the GenPept/GenBank format:

ORIGIN

1 mrqsllﬂts vvpfvlaprp pddpgfgphq rlekldslls dydilslsni

qqhsvrkrdl

61 qtsthvetll tfsalkrhfk lyltssterf sqnfkvvvvd gkneseytvk

wqdfftghvv

121 gepdsrvlah irdddviiri ntdgaeynie plwrfvndtk dkrmlvykse

diknvsrlqs

181 pkvcgylkvd neellpkglv dreppeelvh rvkrradpdp mkntckllvv

adhrfyrymg

241 rgeestttny lihtdran

Note the formatting differences between the GenBank and Fasta sequence formats. Other sequence

formats used for various applications include Raw, DDBJ, Ensembl, ASN.1, Graphics, and XML. Some

of these are output options on the NCBI website, and programs exist (READSEQ; SEQIO) to convert from

one sequence format to another.

542

Basic algorithms of bioinformatics

MATLAB program 9.1

% Pseldotplot.m

% This m-ﬁle compares a protein or DNA sequence with itself, and produces a

% dot plot to display sequence similarities.

% P-selectin protein amino acid sequence pasted directly into m-ﬁle

S=‘MANCQIAILYQRFQRVVFGISQLLCFSALISELTNQKEVAAWTYHYSTKAYSWNISRKYCQN

RYTDLVAIQNKNEIDYLNKVLPYYSSYYWIGIRKNNKTWTWVGTKKALTNEAENWADNEPNNK

RNNEDCVEIYIKSPSAPGKWNDEHCLKKKHALCYTASCQDMSCSKQGECLETIGNYTCSCYPG

FYGPECEYVRECGELELPQHVLMNCSHPLGNFSFNSQCSFHCTDGYQVNGPSKLECLASGIWT

NKPPQCLAAQCPPLKIPERGNMTCLHSAKAFQHQSSCSFSCEEGFALVGPEVVQCTASGVWTA

PAPVCKAVQCQHLEAPSEGTMDCVHPLTAFAYGSSCKFECQPGYRVRGLDMLRCIDSGHWSAP

LPTCEAISCEPLESPVHGSMDCSPSLRAFQYDTNCSFRCAEGFMLRGADIVRCDNLGQWTAPA

PVCQALQCQDLPVPNEARVNCSHPFGAFRYQSVCSFTCNEGLLLVGASVLQCLATGNWNSVPP

ECQAIPCTPLLSPQNGTMTCVQPLGSSSYKSTCQFICDEGYSLSGPERLDCTRSGRWTDSPPM

CEAIKCPELFAPEQGSLDCSDTRGEFNVGSTCHFSCDNGFKLEGPNNVECTTSGRWSATPPTC

KGIASLPTPGVQCPALTTPGQGTMYCRHHPGTFGFNTTCYFGCNAGFTLIGDSTLSCRPSGQW

TAVTPACRAVKCSELHVNKPIAMNCSNLWGNFSYGSICSFHCLEGQLLNGSAQTACQENGHWS

TTVPTCQAGPLTIQEALTYFGGAVASTIGLIMGGTLLALLRKRFRQKDDGKCPLNPHSHLGTY

GVFTNAAFDPSP’;

n=length(S);

ﬁgure(1)

% For ﬁrst ﬁgure, individual residues in the sequence are compared

% against the same sequence. If the residue is a “match”, then a dot is

% plotted into that i, j position in ﬁgure 1.

Figure 9.1

Dot plot output from MATLAB program 9.1. The amino acid sequence of human adhesion protein P-selectin was

obtained from the NCBI Entrez Protein database (www.ncbi.nlm.nih.gov/sites/entrez?db=protein) and compared

against itself while recording single residue matches (a) or two out of three matches (b).

(a)

100 200 300 400 500 600 700 800 900

100

200

300

400

500

600

700

800

900

(b)

100 200 300 400 500 600 700 800 900

100

200

300

400

500

600

700

800

900

543

9.2 Sequence alignment and database searches

for i=1:n

for j=1:n

if S(i)==S(j)

plot(i,j,’.’,‘MarkerSize’,4)

hold on

end

ﬁgure(2)

% For the second ﬁgure, a more stringent matching condition is imposed. To

% produce a dot at position (i, j), an amino acid must match AND one of the

% next two residues must match as well.

for i=1:n-3

for j=1:n-3

if S(i)==S(j) & (S(i+1)==S(j+1) | S(i+2)==S(j+2))

plot(i,j,’.’,‘MarkerSize’,4)

hold on

end

The alignment of two or more sequences can reveal evolutionary relationships and

suggest functions for undescribed genes. For sequences that share a common ancestor,

there are three mechanisms that can account for a character difference at any position:

(1) a mutation replaces one letter for another; (2) an insertion adds one or more letters to

one of the two sequences; or (3) a deletion removes one or more letters from one of the

sequences. In sequence alignment algorithms, insertions and deletions are dealt with by

introducing a gap in the shorter sequence. To determine a numerically “optimal”

alignment between two sequences, one must ﬁrst deﬁne a match score to reward an

aligned pair of identical residues, a mismatch score to penalize an aligned pair of non-

identical residues, and a gap penalty to weigh the placement of gaps at potential

insertion/deletion sites. Box 9.2 shows a simple example of the bookkeeping involved

in scoring proposed sequence alignments. The scoring of gaps can be further reﬁned by

the introduction of origination penalties that penalize the initiation of a gap, and a

length penalty which grows as the length of a gap increases. The proper weighting of

these two scores is used to favor a single multi-residue insertion/deletion event over

multiple single-residue gaps as more likely from an evolutionary point of view.

Alignment algorithms need not weigh each potential residue mismatch as equally

likely. Scoring matrices can be used to incorporate the relative likelihood of different

base or amino acid substitutions based on mutation data. For instance, the tran-

sition transversion matrix (Table 9.2) penalizes an A↔GorC↔T substitution less

severely because transitions of one purine to another purine or one pyrimadine to

another pyrimadine (respectively) is deemed more likely than transversions in which

the basic chemical group is altered.

While common sequence alignment algorithms (e.g. BLAST) for nucleotides utilize

simple scoring matrices such as that in Table 9.2, amino acid sequence alignments utilize

more elaborate scoring matrices based on the relative frequency of substitution rates

observed experimentally. For instance, in the PAM (Point/percent Accepted Mutation)

matrix, one PAM unit represents an average of 1% change in all amino acid positions.

544

Basic algorithms of bioinformatics

ThePAM1matrixwasgeneratedfrom71protein sequence groups with at least 85%

identity (similarity) among the sequences. Mutation probabilities are converted to a log

odds scoring matrix (logarithm of the ratio of the likelihoods), and rescaled into integer

values. Similarly, the PAM250 matrix represents 250 substitutions in 100 amino acids.

Another commonly used scoring matrix is BLOSUM, or “BLOCK Substitution

Matrix.” BLOCKS is a database of about 3000 “blocks” or short, continuous multiple

alignments. The BLOSUM matrix is generated by again calculating the substitution

frequency (relative frequency of a mismatch) and converting this to a log odds score.

The BLOSUM62 matrix (Table 9.3) has 62% similarity, meaning that 62% of residues

Box 9.2 Short sequence alignment

The Kozak consensus sequence is a sequence which occurs on eukaryotic mRNA, consisting of the

consensus GCC, followed by a purine (adenine or guanine), three bases upstream of the start codon

AUG, and then followed by another G. This sequence plays a major role in the initiation of the translation

process. Let’s now consider potential alignments of the Kozak-like sequences in fluit fly, CAAAATG, and

the corresponding sequence in terrestrial plants, AACAATGGC.

Clearly, the best possible alignment

with no insertion/deletion gaps is:

CAAAATG

AACAATGGC (alignment 1)

which shows five nucleotide matches, one C–A mismatch and one A–C mismatch, for a total of two

mismatches. If we place one single-nucleotide gap in the fluit fly sequence to represent a “C” insertion

in the plant sequence, we get

CAA–AATG

AACAATGGC (alignment 2)

which shows six matches, no mismatches, and one gap of length 1. Alternatively, if we introduce a gap

of length 2 in the second sequence, we obtain the following alignment:

CAAAATG

AACA––ATGGC (alignment 3)

which results in five matches, no mismatches, and one gap of length 2. Note that we could have placed

this gap one position to the left or right and we would have obtained the same score. For a match score

of +1, a mismatch score of –1, and a gap penalty of −1, then the three proposed alignments produce

total scores of 3, 5, and 3, respectively. Thus, for this scoring system we conclude that alignment 2

represents the optimal alignment of these two sequences.

Note that we are considering the DNA equivalent of the mRNA sequence, thus the unmethylated

version T (thymine) appears instead of U (uracil).

Table 9.2. Transitio n transversion matrix

ACG T

A 1 −5 −1 −5

C −51−5 −1

G −1 −51−5

T −5 −1 −51

545

9.2 Sequence alignment and database searches

Table 9.3. BLOSUM62 scoring matrix

CSTPAGNDEQHRKMI LVFYW

C9−1 −1 −30−3 −3 −3 −4 −3 −3 −3 −3 −1 −1 −1 −1 −2 −2 −2

S −1 4 1 −1101000−1 −10−1 −2 −2 −2 −2 −2 −3

T −114 1 −1101000−10−1 −2 −2 −2 −2 −2 −3

P −3 −117 −1 −2 −1 −1 −1 −1 −2 −2 −1 −2 −3 −3 −2 −4 −3 −4

A 01−1 −1 4 0 −1 −2 −1 −1 −2 −1 −1 −1 −1 −1 −2 −2 −2 −3

G −301−206 −2 −1 −2 −2 −2 −2 −2 −3 −4 −40−3 −3 −2

N −310−2 −206 100−100−2 −3 −3 −3 −3 −2 −4

D −301−1 −2 −116 20−1 −2 −1 −3 −3 −4 −3 −3 −3 −4

E −400−1 −1 −2025 2001−2 −3 −3 −3 −3 −2 −3

Q −300−1 −1 −20025 0110−3 −2 −2 −3 −1 −2

H −3 −10−2 −2 −211008 0 −1 −2 −3 −3 −2 −12−2

R −3 −1 −1 −2 −1 −20−20105 2 −1 −3 −2 −3 −3 −2 −3

K −300−1 −1 −20−111−125 −1 −3 −2 −3 −3 −2 −3

M −1 −1 −1 −2 −1 −3 −2 −3 −20−2 −1 −1 5 12−20−1 −1

I −1 −2 −2 −3 −1 −4 −3 −3 −3 −3 −3 −3 −314 210−1 −3

L −1 −2 −2 −3 −1 −4 −3 −4 −3 −2 −3 −2 −2224 30−1 −2

V −1 −2 −2 −20−3 −3 −3 −2 −2 −3 −3 −21314 −1 −1 −3

F −2 −2 −2 −4 −2 −3 −3 −3 −3 −3 −1 −3 −3000−1 6 31

Y −2 −2 −2 −3 −2 −3 −2 −3 −2 −12−2 −2 −1 −1 −1 −137 2

W −2 −3 −3 −4 −3 −2 −4 −4 −3 −2 −2 −3 −3 −1 −3 −2 −31211