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

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9.4 End of Chapter 9: key points to consider

(1) 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.

A group of three nucleotides called a “codon”(N =4

= 64 combinations) encodes

the 20 different amino acids that are the building blocks of proteins.

(2) The alignment of two or more sequences can reveal evolutionary relationshi ps and

suggest functions for undescribed genes. For sequences that share a common ances-

tor, 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; (3) a deletion removes one or more letters

from one of the sequences.

(3) When applying a scoring matrix to determine an optimal alignment between two

sequences, a brute-force search of all possible alignments would in most cases be

prohibitive. A powerful and popular algorithm for sequence comparison based on

dynamic programming is called Basic Local Alignment Search Tool,orBLAST.

(4) Some of the speciﬁc BLAST pr ograms available at the NIH National Center for

Biotechnology Information (NCBI, http://blast.ncbi.nlm.nih.gov/Blast.cgi) include:

blastn, for submitting nucleotide queries to a nucleotide database; blastp, for protein

queries to the protein database; blastx, for searching protein databases using a

translated nucleotide query; tblastn, for searching translated nucleotide databases

using a protein query; and tblastx, for searching translated nucleotide databases

using a translated nucleotide query.

(5) There are many situations where it is desirable to perform sequence comparison

between three or more proteins or nucleic acids. Such comparisons can help to

identify functionally important sites, predict protein structure, or reconstruct the

evolutionary history of the gene. A number of algorithms for achieving multiple

sequence alignment are avail able such as ClustalW and T-Coffee.

Note that this tree is unscaled. Similarly, the updated pairwise distance matrix is:

Species (AC)E B

B 14 –

D 9.75 15

As shown in the matrix, D shares more in common with the (AC)E cluster than with species B, and so

our final, unambiguous phylogenetic tree is:

ACEDB

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9.4 Key points

(6) Phylogenetic trees can be us ed to reconstruct genealogies from molecular data, not

just for genes, but also for species of organisms. A phylogenetic tree, or dendrogram,

is an acyclic two-dimensional graph of the evolutionary relationship among three or

more genes or organisms. A phylogenetic tree is composed of nodes and branches

connecting them.

9.5 Problems

9.1. Genome access and multiple sequence alignment Tumor necrosis factor (TNF)-

related apoptosis inducing ligand (TRAIL) is a naturally occurring protein that

speciﬁcally induces apoptosis in many types of cancer cells via binding to “death

receptors.” Obtain the amino acid sequences for the TRAIL protein for human,

mouse, and three other species from the Entrez cross-database search at the National

Center for Biotechnology Information (NCBI) website (www.ncbi.nlm.nih.gov/sites/

gquery). Perform a multiple sequence alignment using the ClustalW and TCoffee

algorithms; these can be accessed from many different online resources. What regions

of the TRAIL protein are most highly conserved between different species?

9.2. Dot plot comparison of two genomic sequences Compare the human TRAIL

protein sequence from Problem 9.1 to the sequence for human tumor necrosis factor

(TNF)-α.TNF-α is an important inﬂammatory cytokine with cytotoxic effects at

sufﬁcient concentration. Generate a dot plot and interp ret the diagram in terms of

any common domains that may be revealed.

9.3. Phylogenetics The following six amino acid sequences are related by a common

ancestor:

EYGINEVV, EVGINAER, EVGINEYR, ENGINRNR, ENLYNEYR, ENGINRYI

Construct a rooted phylogenetic tree for these related sequences, identify the

sequence of the common ancestor, and calculate the distances between the lineages.

References

Altschul, S. F., Madden, T. L., Schaffer, A. A., Zhang, J., Zhang, Z., Miller, W., and

Lipman, D. J. (1997) Gapped BLAST and PSI-BLAST: A New Generation of Protein

Database Search Programs. Nucleic Acids Res., 25, 3389–402.

Carrillo, H. and Lipman, D. (1988) The Multiple Sequence Alignment Problem in Biology.

SIAM J. Appl. Math., 48, 1073–82.

Giasson, B. I., Murray, I. V. J., Trojanowski, J. Q., and Lee, V. M.-Y. (2001) A Hydrophobic

Stretch of 12 Amino Acid Residues in the Middle of a-Synuclein Is Essential for Filament

Assembly. J. Biol. Chem., 276, 2380–6.

Green, C. E., Pearson, D. N., Camphausen, R. T., Staunton, D. E., and Simon, S. I. (2004)

Shear-dependent Capping of L-selectin and P-selectin Glycoprotein Ligand 1 by E-

selectin Signals Activation of High-avidity Beta 2-integrin on Neutrophils. J. Immunol.,

284, C705–17.

Hughey, R. and Krogh, A. (1996) Hidden Markov Models for Sequence Analysis: Extension

and Analysis of the Basic Method. CABIOS, 12, 95–107.

Ivetic, A., Florey, O., Deka, J., Haskard, D. O., Ager, A., and Ridley, A. J. (2004) Mutagenesis

of the Ezrin-Radixin-Moesin Binding Domain of L-selectin Tail Affects Shedding,

Microvillar Positioning, and Leukocyte Tethering. J. Biol. Chem., 279, 33 263–72.

Kim, J., Pramanik, S., and Chung, M. J. (1994) Multiple Sequence Alignment Using

Simulated Annealing. Bioinformatics, 10, 419–26.

558

Basic algorithms of bioinformatics

Needleman, S. B. and Wunsch, C. D. (1970) A General Method Applicable to the Search for

Similarities in the Amino Acid Sequence of Two Proteins. J. Mol. Biol., 48, 443–53.

Notredame, C. and Higgins, D. (1996) SAGA: Sequence Alignment by Genetic Algorithm.

Nucleic Acids Res., 24, 1515–24.

Notredame, C., Higgins, D., and Heringa, J. (2000) T-Coffee: A Novel Method for Multiple

Sequence Alignments. J. Mol. Biol., 302, 205–17.

Sneath, P. H. A. and Sokal, R. R. (1973) Numerical Taxonomy (San Francisco, CA:

W. H. Freeman and Company), pp 230–4.

Thompson, J. D., Higgins, D. G., and Gibson, T. J. (1994) CLUSTAL W: Improving the

Sensibility of Progressive Multiple Sequence Alignment Through Sequence Weighting,

Positions-Speciﬁc Gap Penalties and Weight Matrix Choice. Nucleic Acids Res., 22,

4673–80.

559

References

Appendix A Introduction to MATLAB

MATLAB is a technical computing software that consists of programming

commands, functions, an editor/debugger, and plotting tools for performing math,

modeling, and data analysis. Toolboxes that provide access to a specialized set of

MATLAB functions are available as add-on applications to the MATLAB software.

A Toolbox addresses a particular ﬁeld of applied science or engineering, such as

optimization, statistics, signal processing, neural networks, image processing,

simulation, bioinformatics, parallel computing, partial differential equations, and

several others.

The MATLAB environment i ncludes customizable desktop tools for managing

ﬁles and programming variables. You can minimize, rearrange, and resize the

windows that appear on the MATLAB desktop. The Command Window is where

you can enter commands sequentially at the MATLAB prompt “.” After e ach

command is entered, M ATLAB processes the instructions and displays the results

immediately following the executed statement. A semicolon placed at the end of the

command will suppress MATLAB-genera ted text output (but not graphical

output).

The Command History window is a desktop tool that lists previously run

commands. If you double click on any of these statements, it will be displayed in

the Command Window and executed. The Current Directory browser is another

desktop tool that allows you to view directories, open MATLAB ﬁles, and perform

basic ﬁle operations. The Workspace browser displays the name, data type, and size

of all variables that are deﬁned in the base workspace. The base workspace is a

portion of the memory that contains all variables that can be viewed and

manipulated within the Command Window.

In the Editor/Debugger window you can create, edit, debug, save, and run m-ﬁles;

m-ﬁles are programs that contain code written in the MATLAB language. Multiple

m-ﬁles can be opened and edited simultaneously within the Editor window. The

document bar located at the bottom of the Editor window has as many tabs as there

are open m-ﬁles (documents). Desktop tools (windows) can be docked to and

undocked from the MATLAB desktop. An undocked window can be resized or

placed anywhere on the screen. Undocked tools can be doc ked at any time by

clicking the dock button or using menu commands. See MATLAB help for an

illustration of how to use MATLAB desktop tools.

A1.1 Matrix operations

MATLAB stands for “MATrix LABoratory.” A matrix is a rectangular (two-

dimensional) array. M ATLAB treats each variable as an array. A scalar value is

treated as a 1 × 1 matrix and a vector is viewed as a row or column matrix. (See

Section 2.2 for an introduction to vectors and matrices.) Dimensioning of a variable

is not required in MATLAB. In programming languages, such as Fortran and C, all

arrays must be explicitly dimensioned and initialized. Most of the examples and

problems in this book involve manipulating and operating on matrices , i.e. two-

dimensional arrays. Operations involving multidimensional (three- or higher-

dimensional) data structures are slightly more complex. In this section, we conﬁne

our discussion to matrix operations in MATLAB.

The square brackets [] is the matrix operator used to construct a vector or matrix.

Elements of the matr ix are entered within square brackets. A comma or space

separates one element from the next in the same row. A semicolon separates one

row from the next. Note that the function performed by the semicolon operator

placed inside a set of square brackets is different from that when placed at the end of

a statement . We create a 4 × 3 matrix by entering the following statement at the

prompt:

A = [1 2 3; 4 5 6; 7 8 9; 10 11 12]

12 3

45 6

78 9

10 11 12

MATLAB generates the output immediately below the statement. Section 2.2. 1

demonstrates with examples of how to create a matrix and how to access elements

of a matrix stored in memory. MATLAB provides several functions that create

special matrices. For example, the ones function creates a matrix of ones; the

zeros function creates a matrix of zeros, the eye function creates an identity

matrix, and the diag function creates a diagonal matrix from a vector that is

passed to the function. These functions are also discussed in Section 2.2.1. Other

MATLAB matrix operators that should be committed to memory are listed below.

(1) The MATLAB colon operator (:) represents a series of numbers between the ﬁrst

and last value in the expression (ﬁrst:last). This operator is used to construct a

vector whose elements follow an increasing or decreasing arithmetic sequence; for

example,

a = 1:5;

creates the 1 × 5 vector a = (1, 2, 3, 4, 5). The default increment value is 1. Any other

step value can be speciﬁed as (ﬁ rst:step:last), e.g.

a = 0:0.5:1

0 0.5000 1.0000

Note that use of the matrix constructor operator []with the colon operator

when creating a vector is unnecessary.

(2) The MATLAB transpose operator (')rearranges the elements of a matrix by

reﬂecting the matrix elements across the diagonal. This has the effect of converting

the matrix rows into corresponding columns (or matrix columns into corresponding

rows), or converts a row vector to a column vector and vice versa.

Two or more matrices can be concatenated using the same operators (square

brackets operator and semicolon operator) used to construct matrices. Two matrices

A and B can be concatenated horizontally as follows:

C=[A B];

561

Introduction to MATLAB

Because the result C must be a rectangular array, i.e. a matrix, this operation is

permitted only if both matrices have the same number of rows. To concatenate two

matrices vertically, the semicolon operator is used as demonstrated below:

C = [A; B];

To perform this operation, the two matrices A and B must have the same number of

columns.

Section 2.2.1 shows how to access elements of a matrix. Here, we introduce

another method to extract elements from a matrix. One method is called array

indexing. In this method, a matrix (array) can be used to point to elements within

another matrix (array). More simply stated, the numeric values contained in the

indexing array B determine which elements are chosen from the array A:

A = [1:0.5:5]

1.0000 1.5000 2.0000 2.5000 3.0000 3.5000

4.0000 4.5000 5.0000

B=[23;89];

C = A(B)

1.5000 2.0000

4.5000 5.0000

A special type of array indexing is logical indexing. The indexi ng array B is ﬁrst

generated using a logical operation on A; B will have the same size as A but its

elements will either be ones or zeros. The positions of the ones will correspond to the

positions of those elements in A that satisfy the logical operation, i.e. positions at

which the operation holds true. The positions of the zeros will correspond to the

positions of those elements in A that failed to satisfy the logical operation:

A=[2-13;-1-21];

B=A<0

010

110

Note that B is a logical array. The elements are not of numeric data type (e.g.

double, which is the default numeric data type in MATLAB), but of logical

(non-numeric) data type. The positions of the ones in B will locate the elements in A

that need to be processed:

A(B) = 0

203

001

An empty matrix is any matrix that has either zero rows or zero columns. The

simplest empty matrix is the 0 × 0 matrix, which is created using a set of square

brackets with no elements speciﬁed wi thin, i.e. A=[]is an empty 0 × 0 matrix.

A1.2 Programming in MATLAB

An m-ﬁle contains programming code written in the MATLAB language. All m-ﬁles

have the .m extension. There are two types of m-ﬁles: functions and scripts. A script

562

Appendix A

contains a set of executable and non-executable (commented) statements. It does not

accept input parameters or return output variables. It also does not have its own

designated workspace (memory space). If called from the command line (or run by

clicking the run button from the Editor toolbar), the script directly accesse s the base

workspace. It can create and modify variables in the base workspace. Variables

created by a script persist in the base workspace even after the script has completed

its run. However, if called from within a functi on, a script can access only the

workspace of the calling function.

A function can accept inpu t parame ters and can return output variables to the

calling program. A function begins with a function description line. This line

contains the name of the function, the number of input parameters (or

arguments), and the number of output arguments. One example of a function

statement is

function f = func1(parameter1, parameter2, parameter3)

The ﬁrst line of a function begins with the keyword

function. The name of this

function is func1. A function name must always start with a letter and may also

contain numbers and the underscore character. This function has three input

parameters and returns one output variable f. The name of the m-ﬁle should

be the same as the name of the function. Therefore this function should be saved

with the name func1.m; f is assigned a value within the function and is retur ned by

the function to the calling program. A function does not need an end statement as

the last line of the program code unless it is nested insi de another function. If the

function returns more than one output value, the output variables should be

enclosed within square brackets:

function [f1, f2, f3] = func1(parameter1, parameter2)

If the function does not return any output values, then the starting line of the

function does not include an equals sign, as shown below:

function func1(parameter1, parameter2, parameter3)

If there are no parameters to be passed to the function, simply place a pair of

empty parentheses immediately after the function name:

function f = func1()

To pass a string as an argument to a function, you must enclose the string with

single quotes.

Every function is allotted its own function workspace. Variables created within a

function are placed in this workspace, and are not shared with the base workspace or

with any other function workspaces. Therefore, variables created within a function

cannot be accessed from outside the function. Variables deﬁned within a function are

called local variables. The inpu t and output arguments are not local variables since

they can be accessed by the workspace of the calling program. The scope of any

variable that is deﬁned within a function is said to be local. When the function

execution terminates, local variables are lost from the memory, i.e. the function

A keyword is a reserved word in MATLAB that has a special meaning. You should not use keywords for

any other purpose than what they are intended for. Examples of keywords are if, elseif, else, for,

end, break, switch, case, while, global, and otherwise.

563

Introduction to MATLAB

workspace is emptied. The values of local variables are therefore not passed to

subsequent function calls.

Variables can be shared between a function and the calling program workspace by

passing the variables as arguments into the function and returning them to the

calling function as output arguments.

This is the most secure way of sharing

variables. In this way, a function cannot modify any of the variables in the base

workspace except those values that are returned by the function. Similarly, the

function’ s local variables are not inﬂuenced by the values of the variables existing

in the base workspace.

Another way of extending the scope of a variable created inside a function is to

declare the variable as global. The value of a global variable will persist in the

function workspace and will be made available to all subsequent function calls. All

other functions that declare this variable as global will have access to and be able

to modify the variable. If the same variable is made global in the base workspace,

then the value of this variable is available from the command line, to all scripts, and

to all functions containing statements that declare the variable as global.

Example A1.1

Suppose a variable called data is defined and assigned values either in a script file or at the command

line. (This example is taken from Box 8.1B.)

% Data

global data

time = [0; 1; 3; 5; 10; 20; 40; 50; 60; 90; 120; 150; 180; 210; 240];

AZTconc = [0; 1.4; 4.1; 4.5; 3.5; 3.0; 2.75; 2.65; 2.4; 2.2; 2.15; 2.1;

2.15; 1.8; 2.0];

data = [time, AZTconc];

data is a 15 × 2 matrix. Alternatively, the values of data can be stored in a file and loaded into the

memory using the load command (see Section A1.4).

The function m-file SSEpharmcokineticsofAZTm.m is created to calculate the error in a

proposed nonlinear model. To do this, it must access the values stored in data. Therefore, in this

function, data is declared as global. The first two lines of the function are

function SSE = SSEpharmacokineticsof AZTm(p)

global data

By making two global declarations, one at the command line and one inside the function, the function

can now access the variable data that is stored in the base workspace.

At various places in an m-ﬁle you will want to explain the purpose of the program

and the action of the executable statements. You can create comment lines anywhere

in the code by using the percent sign. All text following % on the same line is marked

as a comment and is ignored by the compiler.

Example A1.2

The following function is written to calculate the sum of the first n natural numbers, where n is an input

parameter. There is one output value, which is the sum calculated by the function. Note the use of

comments.

If the function modiﬁes the input arguments in any way, the updated values will not be reﬂected in the

calling program’s workspace. Any values that are needed from the function should be returned as

output arguments by the function.

564

Appendix A

function s = ﬁrst_n_numbers(n)

% This function calculates the sum of the ﬁrst n natural numbers.

% n: the last number to be added to the sequence

s = n*(n + 1)/2; % s is the desired sum.

In this function there are no local variables created within the function workspace. The input variable n

and the output variable s are accessible from outside the function and their scope is therefore not local.

Suppose we want to know the sum of the first ten natural numbers. To call the function from the

command line, we type

n_sum = ﬁrst_n_numbers(10)

n_sum =

Variables in MATLAB do not need to be declared before use. An assignment

statement can be used to creat e a new variable, e.g. a=2creates the variable a,in

the case that it has not been previously deﬁned. Any variable that is created at the

command prompt is immediately stored in the base workspace.

If a statement in an m-ﬁle is too long to ﬁt on one line in the viewing area of the

Editor, you can continue the statement onto the next line by typing three consecutive

periods on the ﬁrst line and pressing Enter. This is illustrated below (this statement is

in Program 7.13):

f = [-k*y(1)*y(2) + beta*(y(1)+y(3)) – mu*y(1); ...

k*y(1)*y(2) – 1/gamma*y(2); ...

1/gamma*y(2) – mu*y(3)];

Here, f is a 3 × 1 column vector.

A1.2.1 Operators

You are familiar with the arithmetic operators +, −, *, /, and ^ that act on scalar

quantities. In MATLAB these operators can also function as a matrix operator on

vectors and matrices. If A and B are two matrices, then A*B signiﬁes a matrix

multiplication operation; A^2 represents the A*A matrix operation; and A/B

means (right) division

of A by B or A*inv(B). Matrix operations are performed

according to the rules of linear algebra and are discussed in detail in Chapter 2.

Often, we will want to carry out element-by -element operations (array operations)

as opposed to combining the matrices as per linear algebra rules. To perform array

operations on matrices, we include the dot operator, which precedes the arithmetic

operator. The matrices that are c ombined arithmet ically (element-by-element) must

be of the same size. Arithmetic operators for element-by-element operations are

listed below:

.* performs element-by-element multiplication;

./ performs element-by-element division;

.^ performs element-by-element exponentiation.

The statement C = A.*B produces a matrix C such that C(i, j)=A(i, j)*B(i, j). The

sizes of A, B,andC must be the same for this operation to be valid. The statement B=

A.^2 raises each element of A to the power 2.

Left division is explained in Chapter 2.

565

Introduction to MATLAB

Note that + and − are element-wi se operators already and therefore do not need a

preceding dot operator to distinguish between a matrix operation and an array

operation.

For addition, subtraction, multiplication, or division of a matrix A with any scalar

quantity, the arithmet ic operation is identically performed on each element of the

matrix A. The dot operator is not required for specifying scalar–matrix operations.

This is illustrated in the following:

A = [10 20 30; 40 50 60];

B=A– 10

01020

30 40 50

C = 2*B

020 40

60 80 100

MATLAB provides a set of mathe matical functions such as exp, cos, sin,

log, log10, arcsin, etc. When these functions operate on a matrix, the

operation is performed element-wise.

The relational operators available in MATLAB are listed in Table A.1. They can

operate on both scalars and arrays.

Comparison of arrays is done on an element-by-element basis. Two arrays being

compared must be of the same size. If the result of the comparison is true, a “1” is

generated for that element position. If the result of the element-by-element

comparison is false, a “0” is generated. The output of the comparison is an array

of the same size as the arrays being compared. See the example below.

A = [2.5 7 6; 4 7 2; 4 2.2 5];

B = [6 8 2; 4 2 6.3; 5 1 3];

A>=B

ans =

001

010

011

One category of logical operators in MATLAB operates element-wise on logical

arrays, while another category operates only on scalar logical expressions. Note that

logical is a data type and has a value of 1, which stands for true, or has a value of

Table A.1. Relational operators

Operator Deﬁnition

< less than

<= less than or equal to

> greater than

>= greater than or equal to

== equal to

~= not equal to

566

Appendix A