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
Applications of Integration in Biomedical Science
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
Chromatography – separation of mixtures
Types of chromatography used in protein purification:
1.) Ion Exchange
2.) Gel filtration
3.) Hydrophobic
4.) Affinity
Chromatography – separation of mixtures
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)
Types of chromatography – Protein separations
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
Types of chromatography – Protein separations
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.
Types of chromatography – Protein separations
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)
Calculus Topic: Defining area under the curve
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
Answers:Answers:Answers:Answers:
A)A)A)A)
B)B)B)B)
C)C)C)C)
D)D)D)D)
E)E)E)E)
Calculus Topic: Defining area under the curve
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
Answers:Answers:Answers:Answers:
A)A)A)A)
B)B)B)B)
C)C)C)C)
D)D)D)D)
E)E)E)E)
Calculus Topic: Defining area under the curve
Calculus Topic: Defining area under the curve
Which of the three techniques is best?
Why?
Could there be a better way based on your current knowledge of calculus?
Applications of Integration in Biomedical Science
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
Fundamental Theorem of Calculus
Insert clicker question
Integration (Alvaro figure)
Fundamental Theorem of Calculus
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
Methods of detection in chromatography
Methods of detection in chromatography
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
Methods of detection in chromatography
Courtesy Biocompare
DNA absorbs light at approximately 260 nanometers
Methods of detection in chromatography
Proteins absorb light at approximately 280 nanometers
Due to tryptophan and tyrosine residues
Methods of detection in chromatography
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*
Methods of detection in chromatography
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
Methods of detection in chromatography
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
Methods of detection in chromatography
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 – cutting edge use of chromatography
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 – cutting edge use of chromatography
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
Proteomics – cutting edge use of chromatography
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
Proteomics – cutting edge use of chromatography
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.