MA 61 Precalculus for Biology Majors
Lab I: Applications of Linear Functions
Standard curves
A standard curve is a quantitative research tool. It is a method of plotting known assay data to determine
the concentration of an unknown substance, particularly proteins and DNA. The assay is first performed
with various known concentrations of a substance similar to that being measured. A new standard curve
is constructed each time you set up a new series of tests, since we must account for variations in
experimental conditions (solution preparation, spectrophotometer sensitivity, temperature, etc.) The
particular assay procedure used may measure absorbance, optical density, luminescence, fluorescence,
radioactivity, Rf or something else.
This data obtained from a set of known samples is then used to make a standard curve, by plotting the
concentration on the x-axis, and assay measurement on the y-axis. If the data exhibits a linear
relationship the standard curve will be a straight line. The same assay is then performed with samples of
unknown concentration. To analyze the data, one locates the measurement on the y-axis that corresponds
to the assay measurement of the unknown substance and follows a line to intersect the standard curve.
The corresponding value on the x-axis is the concentration of substance in the unknown sample.
A line in a two variables is defined by the equation y = mx + b. The y variable can be expressed in terms
of a constant (b) and a slope (m) times the x variable. The constant is also referred to as the intercept (or
more specifically the y-intercept), and the slope as the regression coefficient. A correlation coefficient
(R) is also computed. R is a quantity that gives the quality of a least squares fitting to the original data.
The closer R is to 1 the better the approximation. Typically an R-value greater than 0.97 (or an R2 value
of 0.95 or higher) is needed for a good linear approximation.
In this lab we will learn how to create standard curves using Excel and then use them to determine the
concentration level of unknown samples. We will also show how to find the associated line
algebraically. The focus will be on linear functions or graphs that can be turned into linear functions by
a scale (axis) change. We will accomplish this by learning how to:
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

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Enter Data into an Excel Spreadsheet
Create and Modify Graphs
o Using Chart Wizard to Plot the Data
 Selecting Chart Type
 Source Data
 Chart Options
 Location
 Add Data
 Format Graph
Find a Standard Curve Graphically and Algebraically
Create Trendlines to find Standard Curves in Excel
o Display equation
o Display r2 value
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Lab Assignment Part 1: Standard Curve for Protein Measurements:
A standard curve for protein concentration is often created using known concentrations of bovine serum
albumin. The assay is called the Bradford assay; it is a colorimetric assay. The reagent turns blue when it
binds to amino acids present in protein. The intensity of the color is best measured with a
spectrophotometer (A device for comparing two light radiations, wavelength by wavelength). In the case
of the Bradford assay the greater the absorbance, the higher the protein concentration.
A series of tests were performed on some samples and the following measurements were obtained using
a spectrophotometer:
Protein
Concentration
(mg/ml)
0.28
0.56
0.84
1.12
1.40
Absorbance (A)
0.078
0.143
0.393
0.473
0.527
TASKS:
1. Enter this data into Excel
2. Create a graph of the data that is appropriately titled and labeled (Include your name on the
graph)
3. Print out the graph
4. Use a straight edge to sketch a line that “best” approximates the trend of this data.
5. Estimate the slope and y-intercept and use them to find the equation of the associated line using
the slope-intercept form of a line: y = mx + b
6. Pick two points on the line that you’ve drawn [(x1,y1) and (x2,y2)] and use these points to find the
equation of this line using the two-point formula for a line:
 y  y1 
y  y1   2
  x  x1 
 x 2  x1 
Solve the equation for y.
7. Now use the Trendline feature of Excel to find the “best” fit line to the data.
8. Display the equation and the R2 value of the trendline on the graph and print out the graph.
9. Compare your algebraically derived equations to the Excel generated equation.
10. Use the Excel equation to determine the protein concentrations of the following unknown
samples:
Absorbance (A)
0.023
0.223
0.201
0.642
0.760
Show your work.
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Lab Assignment Part 2: Standard Curve for DNA Measurements:
Pure, dissolved DNA forms a transparent, colorless solution. DNA will react with a chemical called
diphenylamine (DPA), and the reaction produces a blue color. The DPA reacts with the deoxyribose in
the DNA and is quantitative: that is the more DNA there is, the more intense the blue color. The
intensity of the blue color is measured using a spectrophotometer as described in Assignment 1 above.
A series of tests were performed and the following measurements were obtained using a
spectrophotometer:
g DNA in
the Sample
0
100
200
400
600
800
1000
Absorbance (A)
0
0.064
0.132
0.253
0.415
0.512
0.694
TASKS:
1. Enter this data into Excel
2. Create a graph of the data that is appropriately titled and labeled (Include your name on the
graph)
3. Print out the graph
4. Use a straight edge to sketch a line that “best” approximates the trend of this data.
5. Estimate the slope and y-intercept and use them to find the equation of the associated line using
the slope-intercept form of a line: y = mx + b
6. Pick two points on the line that you’ve drawn [(x1,y1) and (x2,y2)] and use these points to find the
equation of this line using the two-point formula for a line:
 y  y1 
y  y1   2
  x  x1 
 x 2  x1 
Solve the equation for y.
7. Now use the Trendline feature of Excel to find the “best” fit line to the data.
8. Display the equation and the R2 value of the trendline on the graph and print out the graph.
9. Compare your algebraically derived equations to the Excel generated equation.
10. Use the Excel equation to determine the DNA concentrations of the following unknown samples:
Absorbance (A)
0.015
0.280
0.356
0.480
0.763
Show your work.
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Lab Assignment Part 3: Standard Curve for Molecular Weight Measurements:
Determination of molecular weight is sometimes accomplished by using a relative mobility
measurement technique. A common abbreviation for relative mobility is Rf (the f is a subscript), for
"retention factor." This procedure is called electrophoresis and involves the relative mobility (the
distance migrated by a band divided by the distance migrated by the dye front) of a substance being
tested. Relative mobility of a polypeptide band is related to its molecular mass. By running a set of
protein standards of known molecular mass, a standard curve of mass versus relative mobility can be
produced, from which estimates of apparent mass can be made for unknown proteins. The Rf is
calculated as the ratio of the distance migrated by the molecule to that migrated by a marker dye-front.
See picture below.
A simple way of determining relative molecular weight by electrophoresis is to plot a standard curve of
Rf vs. log(Mw) for known samples, and read off the log(Mw) of the sample after measuring distance
migrated on the same gel. After introducing the data of ladder and sample Rf, sample Mw is calculated.
A linear relationship exists between the logarithm of the molecular weight (Mw) of the nucleic acid, and
its Rf.
Log(Mw) = m Rf + b.
This is and example of a log-linear relation. The relation is linear in one variable (Rf) and logarithmic in
another (Mw). In this section we show how to work we these relations in two different ways.
A series of tests were performed and the following Rf measurements were obtained:
Molecular Wt.
15848.93
39810.72
63095.73
125892.5
Log (Mw)
4.2
4.6
4.8
5.1
Rf
0.37
0.27
0.24
0.16
TASKS:
1. Enter this data into Excel
2. Create two graphs of the data that are appropriately titled and labeled (Include your name on the
graphs) One graph should use the actual molecular weight as the independent variable while the
other graph should use the logarithm of the molecular weight as the independent variable.
3. Print out both graphs
4. Use a straight edge to sketch a line that “best” approximates the trend of these data.
5. Pick two points on each line that you’ve drawn [(x1,y1) and (x2,y2)] and use these points to find
the equations of these lines using the two-point formula for a line:
 y  y1 
y  y1   2
  x  x1 
 x 2  x1 
Solve the equation for y.
6. Now use the Trendline feature of Excel to find the “best” fit lines to the data.
7. Display the equations and the associated R2 values of the trendlines on the graphs and print out
the graphs.
8. Using the graph that displays the actual molecular weight, change the axis to display logarithmic
values instead.
9. Print this graph out
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10. Use the Excel trendline feature and construct a logarithmic trendline and print out both the
equation and the R2 value and print the resulting graph.
11. Compare your equations to the Excel generated equations.
12. Discuss the difference between the trendline from the graph with the actual molecular data
displayed on a logarithmic scale with that of the graph using the logarithm of the data.
13. Use one of the Excel equations (your choice) to determine the molecular weights of the
following unknown Rf samples:
Rf
0.43
0.31
0.17
0.06
Show your work.
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