applications of integration in biomedical science · pdf fileapplications of integration in...

Applications of Integration in Biomedical Science

by

William T. Self

UCF EXCEL Applications of Calculus

Calculus Topic: Defining area under the curve

Topic #1: Approximating rectangles Topic #1: Approximating rectangles Topic #1: Approximating rectangles Topic #1: Approximating rectangles

One possible method for estimating One possible method for estimating One possible method for estimating One possible method for estimating area under a given curve (or area under a given curve (or area under a given curve (or area under a given curve (or function) is the use of approximating function) is the use of approximating function) is the use of approximating function) is the use of approximating rectanglesrectanglesrectanglesrectangles

This is a simple method, but has This is a simple method, but has This is a simple method, but has This is a simple method, but has limitations in its ability to accurately limitations in its ability to accurately limitations in its ability to accurately limitations in its ability to accurately define the areadefine the areadefine the areadefine the area

Calculus concept # 1

Section 5.1 #1: Section 5.1 #1: Section 5.1 #1: Section 5.1 #1: By reading values from the given graph of f(shown on the next slide) use threethreethreethree rectangles to find a lowerowerowerower estimate for the area under the given graph of f from x=0 to x=6. In each case sketch the rectangles that you use.

Approximating Rectangles

0 1 2 3 4 5 6 7 8 9 10 110

1

2

3

4

5

6

7

8

x

yy = f(x)

Reminder:

3 rectangles

Lower limit

From x=0 to x=6

Approximating rectangles

Answers:

A) 17

B) 19

C) 21

D) 20

E) 28

Approximating rectangles

The use of this technique is inadequate to determine the area under a curve since it can overestimate and underestimate this area

This section of the applications course will introduce you to concepts and methods in biomedical science that rely on calculus to determine the quantity of compounds and macromolecules

Some of the future courses (that you may take) that this will be relevant:

MCB 3020 – General Microbiology

BSC 3403C – Quantitative Biological Methods

MCB 4414 – Microbial Metabolism

BCH 4053 – Biochemistry I

BCH 4054 – Biochemistry II


Life – Its existence on Earth

Time Line for Planet Earth

ProkaryotesProkaryotes

EukaryotesEukaryotes

Prokaryotes

� involved in formation of the biosphere

� required for plant & animal survival

eukaryotic cell prokaryotic cell

Life – Cellular level

What are cells made of (E.coli )?

CHNOPS:Carbon

Hydrogen

Nitrogen

Oxygen

Phosphorus

Sulfur

Adenosine triphosphate - ATP

Biological Macromolecules

Trace ElementsHuman composition (complements of Dept. of Energy)Human composition (complements of Dept. of Energy)Human composition (complements of Dept. of Energy)Human composition (complements of Dept. of Energy)

Dry weight %Dry weight %Dry weight %Dry weight %Carbon 61.7Nitrogen 11.0Oxygen 9.3Hydrogen 5.7Calcium 5.0Phosphorus 3.3Potassium 1.3Sulfur 1.0Chlorine 0.7Sodium 0.7Magnesium 0.3

Trace amounts of B, F, Si, V, Cr, Mn, Fe, Co, Cu, Zn, Se, Mo, STrace amounts of B, F, Si, V, Cr, Mn, Fe, Co, Cu, Zn, Se, Mo, STrace amounts of B, F, Si, V, Cr, Mn, Fe, Co, Cu, Zn, Se, Mo, STrace amounts of B, F, Si, V, Cr, Mn, Fe, Co, Cu, Zn, Se, Mo, Sn, I.n, I.n, I.n, I.There are some arguments as to the importance of other trace eleThere are some arguments as to the importance of other trace eleThere are some arguments as to the importance of other trace eleThere are some arguments as to the importance of other trace elementsmentsmentsments

Biological Cells – Complex mixtures

Basics:

DNA, RNA: Polymers of nucleic acids – encode proteins

Proteins: Polymers of amino acids – can be structural or act as enzymes

Lipids: Polymers of carbon – structural components of cell membranes

A given cell will have thousands of different proteins, RNA molecules and metabolites present under a particular growth condition

How do we define the ‘role’ of each individual protein (for example)?

First we must purify this protein away from all other components, then study it in a test tube (in vitro)

Biological Cells – Complex mixtures

Protein Purification

Proteins are polymers of amino acids

Protein sequence defines the chemical composition

Each protein has unique size, charge and shape

Chromatography – separation of mixtures

Chromatography in general is the separation of compounds from mixtures using a Solid Solid Solid Solid phasephasephasephase and a mobile phasemobile phasemobile phasemobile phase

Typically the solid phase is stationary, and held in place in a column

The mobile phase (usually aqueous) moves through the solid phase and carries the sample

Samples separate from each other on the column due to differences in their unique properties:

1.) net charge

2.) hydrophobicity

3.) size

4.) specific affinity


Types of chromatography used in protein purification:

1.) Ion Exchange

2.) Gel filtration

3.) Hydrophobic

4.) Affinity


Types of chromatography – Protein separations

1.) Ion exchange:

The solid phase has a strong or weak charged group (e.g. strong positive charge)

If a protein has a net negative charge (anionic), it will bind to a column that has a cationic group (positive charge). Each protein will have a slightly different net charge and thus mixtures of proteins can be separated based on net charge

2.) Gel filtration

Proteins will separate based on size, due to pores present in beads in the solid phase. The pores define the separation capabilities of the media (e.g. 30,000 MW to 3,000,000 MW)


3.) Hydrophobic Interaction Chromatography

Based on binding of hydrophobic amino acids (such as leucine, isoleucine) that are usually buried but occasionally present on the surface

Common groups on the stationary phase are phenyl groups or carbon chains


4.) Affinity ChromatographyGenerally, proteins can be engineered to contain ‘tags’ at their ends that will bind to a certain group (e.g. metal). This tag is usually unique in the mixture and thus a ‘tagged’protein can be purified quite readily from a cell extract using this procedure.

The use of protein tags has revolutionized the study of proteins in enzymes in the wake of the era of molecular biology and cloning.


How does this relate to Calculus???

To find and determine the quantity of a given protein, or other molecule of interest, we follow the elution of these molecules using a detector. This pattern is essentially a continuous function from one time period to the next as follows:

Samples eluting become a series of peaks that can be followed and quantified by area under the curve

Calculus concept # 2

Section 5.1 #2 Use 4 rectangles to find Section 5.1 #2 Use 4 rectangles to find Section 5.1 #2 Use 4 rectangles to find Section 5.1 #2 Use 4 rectangles to find estimates of each type for the area under the estimates of each type for the area under the estimates of each type for the area under the estimates of each type for the area under the given graph of given graph of given graph of given graph of ffff from from from from xxxx = 0 to = 0 to = 0 to = 0 to xxxx =6.=6.=6.=6.

Three questions Three questions Three questions Three questions –––– left and right endpoints and left and right endpoints and left and right endpoints and left and right endpoints and finally midpointsfinally midpointsfinally midpointsfinally midpoints

Limitations of approximating rectangles

0 2 4 6 8 10 12−1

0

1

2

3

4

5

6

7

8

9

10

y = f(x)

x

y

Reminder:

Left Left Left Left endpoints 3

Answers:Answers:Answers:Answers:

A)A)A)A)

B)B)B)B)

C)C)C)C)

D)D)D)D)

E)E)E)E)


0 2 4 6 8 10 12−1

0

1

2

3

4

5

6

7

8

9

10

y = f(x)

x

y

Reminder:

Right Right Right Right endpoints 3


A)A)A)A)

B)B)B)B)

C)C)C)C)

D)D)D)D)

E)E)E)E)


0 2 4 6 8 10 12−1

0

1

2

3

4

5

6

7

8

9

10

y = f(x)

x

y

Reminder:

Midpoints 3Midpoints 3Midpoints 3Midpoints 3


A)A)A)A)

B)B)B)B)

C)C)C)C)

D)D)D)D)

E)E)E)E)



Which of the three techniques is best?

Why?

Could there be a better way based on your current knowledge of calculus?


In addition to protein purification, chromatography (and area under the curve) has many other uses in biomedical science

Some issues to discuss:

1.) Arsenic (and other contaminants) in drinking water

2.) Drug testing (e.g. steroid use)

3.) Bioterrorism – detection of explosives

4.) Pesticides in agriculture and consumer use

Applications of chromatography

How do we determine the presence of a pesticide present in a lake, river or stream?

How do we quantify such a compound?

Why does this quantization matter?

Drug testing – front lines

Websites for discussion:

http://www.questdiagnostics.com/employersolutions/standard_urine_testing_es.html

http://www.agilent.com/about/newsroom/lsca/background/2007/bg_sports_drug_testing.pdf

Drug testing – front lines

Recent article in the journal Nature outlines issues in drug testing for anabolic steroids

Example of LC-MS analysis

Overview of typical HPLC setup:

Detector is typically a mass spectrometer that can predict the mass of eluting compounds

Example of LC-MS analysis

Agilent Technologies example of LC profile of steroids

Gas chromatography (GC)

Gas chromatography:Similar to HPLC, with the exception that the mobile phase is a gas

Sample is either a gas or is derivatized to a volatile form to allow for separation in a gas mobile phase

Column has a liquid stationary phase which is bound to an inert support phase that is solid

This form of chromatography is most common in analytical analysis of pesticides and lipid analysis.

GC – typical set up

Typical GC setup – courtesy of Waters, Inc.

Explosives – GC-MS example

Small amounts of explosives can be buried in compounds that ‘mask’ their presence in samples

GC-MS can uncover readily

Calculus concept #3

Fundamental theorem of calculusThe fundamental theorem of calculus states: (Part 1)

g(x) = ∫ f (t )dt

where f is a continuous function on [ a,b] and x varies between a

and b.

y = f(t)

y

xbxa

area = g(x)

Fundamental Theorem of Calculus

Fundamental Theorem of CalculusPart 2 states:

If f is continuous on [a,b] then:

∫ f (x ) dx = F (b ) – F (a )

Essentially, for purposes of defining area under the curve, the difference in the antiderivative of f between two points [a,b ] on the curve (assuming a continuous function) is equal to the area of that curve to the x-axis

This is the most critical application (in biological sciences) of the fundamental theorem


Insert clicker question

Integration (Alvaro figure)

Biomedical Science - review

What are the three most abundant elements in the human body (dry weight analysis)?

A.) Hydrogen, Nitrogen and Calcium

B.) Carbon, Nitrogen and Hydrogen

C.) Magnesium, Carbon and Nitrogen

D.) Carbon, Oxygen and Hydrogen

E.) Carbon, Selenium and Magnesium

Methods of detection in chromatography

After separation (HPLC, GC, etc.) we must identify and quantify a molecule of interest

Some of the commons ways to find and quantify molecules:

1.) UV-visible spectroscopy2.) Mass spectrometry3.) Flame ionization (FID)4.) Thermal conductivity (TCD)

These abbreviations lead to the multitude of common analytical techniques:

LC-MS (Liquid chromatography – detection by mass spectrometry

GC-MS, etc.

All are still based on the fundamental concepts of chromatography, and all can use integration of peak area to determine the quantity of an eluted sample



1.) UV1.) UV1.) UV1.) UV----visible spectroscopyvisible spectroscopyvisible spectroscopyvisible spectroscopy

Functional groups in a molecule can absorb light at a given wavelength

Aromatics and metal-complex ligands are common groups in biological samples that absorb light in UV or visible range


Courtesy Biocompare

DNA absorbs light at approximately 260 nanometers


Proteins absorb light at approximately 280 nanometers

Due to tryptophan and tyrosine residues


HPLC analysis of purines

A purine metabolizing enzyme was tested for its substrate specificity (which compounds it acts on) using HPLC analysis

Each substrate and product elutes at a different time from reverse phase HPLC (hydrophobic stationary phase)

Purines followed by UV-vis*


2.) Mass spectrometry2.) Mass spectrometry2.) Mass spectrometry2.) Mass spectrometry

Mass spectrometry determines the overall predicted molecular weight of a molecule based ‘weighing’ its charge to mass ratio

Molecules are charged in an ion source, then accelerated to a high speed. They are then passed through a magnetic field and their trajectory is altered by this field, dependent on their charge to mass ratio


Image courtesy of USGS

The particles are then detected and their composition can be predicted based on this charge to mass ratio

Other information on the sample is generally needed to be able to identify and confirm the molecule of interest


3.) Flame Ionization Detection (FID)3.) Flame Ionization Detection (FID)3.) Flame Ionization Detection (FID)3.) Flame Ionization Detection (FID)

FID is commonly used in GC applications, and is based on ‘burning’ of the sample

FID is very good at detecting hydrocarbons and other carbon containing molecules

4.) Thermal Conductivity Detection (TCD)4.) Thermal Conductivity Detection (TCD)4.) Thermal Conductivity Detection (TCD)4.) Thermal Conductivity Detection (TCD)

TCD is commonly used to detect gases (hydrogen) when carried in an inert gas (argon)

TCD is based on changes in thermal conductivity –useful since it can detect nearly any compound

Use of calculus in Biomedical Science - Review

What characteristic of proteins is useful in gel filtration chromatography?

A.) Affinity for ligands

B.) Net charge

C.) Hydrophobicity

D.) Size

E.) Sequence

Proteomics – cutting edge use of chromatography

Cancer diagnosis: Current techniques

Example: Breast cancer

Mammogram

Ultrasound

Biopsy

Genetic screening

Expensive, labor intensive and usually only detect cancer at later stages (not when first forming)


Proteomics:Proteomics:Proteomics:Proteomics:The proteome is defined as the set of proteins present in the cell under a given growth condition

The complement of proteins changes in different cell types (tissues) and under different conditions (stress, infection, disease)

Genetic variability also is displayed in the proteome


Proteomics in Cancer diagnosis:

Using reverse phase chromatography to follow the ‘proteome’ of a clinical sample (e.g. serum), one can obtain a profile of the peptides that are present in a patient

Analysis of hundreds of patients, both ill and healthy, allow for patterns to emerge in this analysis


Above is a sample chromatogram of the peptides in serum of an ovarian cancer patient

Biomarkers of Ovarian Cancer, Gynecologic Oncology 88, S25–S28 (2003)

doi:10.1006/gyno.2002.6679


Proteomic analysis to diagnose cancer:

In a study published in 2002 using a blinded set of samples, the proteomic pattern correctly predicted 36 (95%, 95% confidence interval [CI] = 82% to 99%) of 38 patients with prostate cancer, while 177 (78%, 95% CI = 72% to 83%) of 228 patients were correctly classified as having benign conditions.

Serum proteomic patterns for detection of prostate cancer.

J Natl Cancer Inst. 2002 Oct 16;94(20):1576-8.

applications of integration in biomedical science · pdf fileapplications of integration in...

Documents