Download Lab 1 Scientific Experimentation: Standard Curve Analysis

Scientific Experimentation: Standard Curve Analysis
for DNA Quantitation
Introduction
The Scientific Process is a method by which humans systematically ask
questions of nature in order to understand how things work. It is based on the
idea that nature works according to regular repeating rules and that by careful,
systematic observation, we can discover those rules. The ideas of science are
that humans can find things out directly from experience without having to
depend on other humans (or books, etc.) for knowledge, and that the rules that
are deduced can be used to make predictions about the outcome of future events
so we can plan effective actions. Scientists write down the conclusions they
reach so that other people can benefit from them without having to do every
experiment personally. However, they must always report the experimental
methods and evidence from which the conclusions were drawn so that others
can repeat the experiments or independently evaluate the evidence. Scientific
principles that have received wide acceptance have been tested in many ways,
from many sides, by many people and have been debated and discussed until
everyone is quite sure that the evidence all supports the principle as stated.
Never would a major principle be based on the work of one person. If later
evidence appears that is in conflict with even the most highly respected scientific
principle, the principle would have to be restated to reflect the new knowledge.
Science is limited in scope: if some things do not function according to rules, or if
the rules involve too many variables in too many types of interactions for the
human mind to unravel, then those things cannot be solved by the scientific
method—at least for now. It does no good to ask nature for the name of the
capital of Nebraska. Humans chose the name in an arbitrary manner; it does not
obey a rule of nature. Nor have humans yet successfully unraveled all the
variables involved in human interactions so no firm predictions can be made
about how political groups such as Congress will behave. Why do some works
of art move many people deeply while others do not? There are many important
spheres of human life, including value systems and emotions, which cannot be
tested or predicted by science.
It is important for modern humans to understand which parts of our lives can be
enriched by science and which require other skills and knowledge from us. As
complete human beings who make decisions and choices about our lives,
including ethical and moral ones, we need to use scientific knowledge and the
scientific process in appropriate coordination with the skills and values of other
disciplines.
In the scientific method, facts about nature are obtained through direct
observation and measurement of events, usually under controlled conditions that
comprise an experiment. Then these facts are interpreted and analyzed to obtain
Standard Curve Analysis for DNA Quantitation
Mary Colavito and Ruth Logan, 2010
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an understanding of the rules that controlled the observed events. Scientists are
always careful to separate facts (direct observations and measurements) from
interpretation, and to always show how interpretation is related to and derived
from facts.
In general, all facts must be used or incorporated into an interpretation since to
ignore some might involve omitting important information that has a bearing on
the rules that are being sought. All facts contain unplanned variables or
conditions (error) that will cause the scientist to make an inappropriate
conclusion if not discovered, so part of every analysis involves trying to expose
such variables in order to reveal and remove their causes if possible and to
correct the analysis of the phenomenon for their effects.
Scientists finally draw a conclusion about each experiment or step that they
complete in discovering rules of nature. These usually represent a restatement of
a hypothesis that guided their initial experimental design. In their simplest form,
hypothesis and conclusion statements are similar and represent a precisely
described relationship between two measurable variables. It is this form that we
will practice and use.
You will have opportunities to learn about and practice the scientific process
throughout this semester and the next two (Biology 22 and 23) since full
understanding of where scientific information comes from and confident use of
the process are very important to anyone becoming a scientist.
In this first laboratory session, you will gain experience with measurements and
methods that enhance the accuracy of experimentation. The goals of this
exercise are to:
1. Separate the cellular components of a plant cell to obtain plant DNA.
2. Develop a consistent process for measuring the concentration of a
component in solution.
3. Process data to make a standard curve graph. Graphs are pictorial
representations of the process studied in the experiment.
4. Use the standard curve graph to determine the concentration of the
unknown DNA solution.
5. Use collective standard curve data to evaluate the variability that occurs
during experimentation.
6. Formulate a laboratory report that presents the data and your
interpretation of the data.
The exercise is divided into three parts, which should be completed in the
following order:
A. Strawberry DNA Extraction
B. Standard Curve Preparation
C. DNA Quantitation
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Materials used in this laboratory are non-hazardous. Solids can be discarded in
the trash basket. To insure optimum safety, blue-stained solutions should be
poured into the chemical waste container. All other solutions can be safely
poured down the sink drains.
A. STRAWBERRY DNA EXTRACTION
Each student should perform the DNA isolation individually.
1. Place one strawberry in a zip-loc bag and seal the top of the bag
completely.
2. Mash the strawberry with your hands for about 2 minutes.
3. Add 10 ml of DNA Extraction Buffer.
This buffer contains detergent and salt (sodium chloride). How do
these two components affect the strawberry cells?
4. Reseal bag and mash the strawberry again for 2 minutes.
5. Line a plastic funnel with cheesecloth. Place the funnel in a 150 ml
beaker.
6. Cut a corner from the zip-loc bag and empty the strawberry solution into
the cheesecloth.
7. Squeeze the solution into the funnel and allow it to collect in the beaker.
There may also be some foam but this does not interfere with obtaining
the DNA. Discard the cheesecloth along with the pulp that remained
inside.
This step recovers the cellular material that is soluble in the
extraction buffer.
8. Transfer one-third to one-half of the strawberry solution to a 15 ml tube.
9. Using a plastic transfer pipette, slowly add ice-cold ethanol down the side
of the tube so that it forms a layer on top of the strawberry solution.
Continue adding the ethanol until the tube is about three-quarters filled
and you see the DNA appearing as a white, wispy material at the interface
between the strawberry solution and ethanol.
What is the effect of ethanol on the DNA?
10. Place a wooden stick into the tube at the solution-ethanol interface and
rotate gently to wind the DNA around the stick.
11. Add 0.5 ml of 70% ethanol to a 1.5 ml microcentrifuge tube. Transfer the
DNA from the stick into this solution. The material will be sticky and you
may need to use the transfer pipette to push the DNA from the stick. Use
an additional 0.5 ml of 70% ethanol to completely transfer the DNA,
ending with a total volume of 1 ml in the microcentrifuge tube.
12. Collect the DNA by spinning the tube in a microcentrifuge for 1 minute at
high speed. For this and all subsequent centrifugations, follow your
instructor’s directions for loading, balancing and operating the
microcentrifuge and be sure to accommodate the tubes of your
classmates.
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13. At the end of centrifugation, the ethanol solution will be the liquid
supernatant and the DNA will be found in a solid pellet on the side of the
tube near the bottom. Remove as much of the ethanol as possible without
disturbing the DNA pellet. If necessary, use a transfer pipette or a tissue to
take the residual ethanol from the tube.
14. Let the tube air dry for 5 minutes.
15. Add 5 drops of 0.2% methylene blue to the DNA pellet. Use the wooden
stick or a transfer pipette to mix the dye with the DNA. Allow this mixture
to incubate at room temperature for 5 minutes.
Methylene blue is a dye that binds to DNA.
16. Using a transfer pipette, fill the microcentrifuge tube with tap water. Close
the tube and mix the contents by inverting the tube. Spin the tube in the
microcentrifuge at high speed for 1 minute.
Water is used to remove excess methylene blue dye that is not
bound to DNA.
17. Remove as much of the water as possible without disturbing the DNA
pellet. Refill the microcentrifuge tube with tap water and repeat the mixing
step. Spin the tube in the microcentrifuge at high speed for 1 minute.
18. Remove as much of the water as possible without disturbing the DNA
pellet. Using a transfer pipette, resuspend the DNA in 1 ml tap water.
Transfer to a small glass cuvette and add 4 ml of tap water. Uniformly
resuspend the blue-stained DNA in the 5 ml solution by pipetting up and
down repeatedly.
This treatment will break the long DNA strands somewhat, but
will still allow detection of the amount of DNA recovered.
19. Your stained DNA solution is highly stable and should be saved for the
remainder of the laboratory period. As part of the standard curve
analysis, read and record the absorbance of the 5 ml DNA-containing
solution at 550 nm.
You will use this value along with your standard curve to determine
the concentration of DNA recovered from a single strawberry.
20. Clear your work area in preparation for the next part of the experiment.
Rinse all used glassware and invert on a paper towel to dry. Discard all
solids in the trash. Pour any blue-stained solutions in the chemical waste
bucket. Wipe any spilled liquids from the bench top.
B. STANDARD CURVE PREPARATION
A standard curve is a common experimental tool used to determine the
concentration of a solution. Light absorbed by a series of known concentrations
of the solution is measured and a graph is produced that relates absorbance to
concentration. The graph can be used to extrapolate the concentration of an
unknown solution once its absorbance is obtained. For this part of the exercise,
you will systematically dilute a concentrated solution of methylene blue and read
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the absorbance of each solution using a spectrophotometer. This data will be
plotted for a standard curve. You will also measure the absorbance of your
stained DNA solution and use the standard curve to determine the yield of DNA
that you obtained from a single strawberry. In addition, standard curve data from
all laboratory groups will be combined and analyzed to better understand the
variability that arises during scientific experimentation. The instructions for
producing the standard curve follow the introduction to the spectrophotometer,
the instrument used to measure light absorbance.
INTRODUCTION TO THE SPECTROPHOTOMETER
The spectrophotometer is an instrument used to measure the amount of light that
can pass through a solution. This, in turn, can be used as a measure of the
concentration of a substance in the solution because of Beer’s Law which states
that the concentration of a substance in solution is directly proportional to the
amount of light absorbed by the solution, and inversely proportional to the
logarithm of the fraction of light transmitted by the solution.
Absorbance = log10 Io = as x Concentration I
Where: Io = incident light intensity
I = final light intensity
as = extinction coefficient (a constant)
% Transmittance
Absorbance
Concentration (mg/mL)
Concentration (mg/mL)
Beer’s law only holds if the “incident light” (the light which enters the solution) is
monochromatic, that is, composed of light of a single wavelength. Normal white
light is a mixture of many different wavelengths between 380nm and 750nm
(nanometer = 10-9 meter). Our eyes and brain interpret these different
wavelengths as different colors. Some approximate equivalencies between
wavelength and color are given below:
400-435 nm Violet
435-480 nm Blue
480-580 nm Green
580-595 nm Yellow
5
595-610 nm Orange
610-750 nm Red
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The spectrophotometer can separate white light into its component wavelengths
by means of a prism (diffraction grating). The operator of the instrument can
select incident light of any wavelength by turning the dial that rotates this prism.
The light enters a cuvette that contains the test solution and part of it will be
absorbed if the substance in the solution is “active” (that is, chemically able to
absorb light) at this wavelength. Unabsorbed light (transmitted light) strikes a
photocell on the other side of the cuvette and generates an electric current which
is registered on a galvanometer scale. The operator of the spectrophotometer
can read the “percent transmittance” (%T) or the absorbance (Abs) directly on
the meter.
Since the amount of light absorbed by a solution is directly proportional to the
concentration of that solution, while the percent transmittance is related to the
concentration as an inverse logarithmic function, we usually use the absorbance
scale rather than the percent transmittance scale to determine the concentration
of a solution. This makes the determination much easier than it would be
otherwise: for example, if the concentration of an unknown solution were half that
of a known solution, its absorbance would also be half that of the known solution.
Since the absorbance of an unknown solution is directly proportional to its
concentration, we can easily determine its concentration by comparing its
absorbance with the absorbance of a solution of known concentration.
Absorbance(Abs) (known) =
Absorbance (unknown)
Concentration (known)
Concentration (unknown)
Therefore:
Concentration (unknown) = Concentration (known) X Absorbance (unknown)
Absorbance (known)
Since all experimental techniques are subject to variability (error), it is usual to
find the absorbance of a particular solution more than once, then to use a
statistical method to find a probable value for the measurement.
A standard curve, showing the relationship between absorbance and
concentration, can be prepared for any substance at any specific wavelength.
This can then be used to determine the concentration of unknown solutions in the
future. An example of a standard curve and its use are given on the next page.
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An example of a standard curve using Potassium Permanganate dissolved in water.
Dilutions of a solution containing potassium permanganate were prepared and their concentrations
calculated from that of the original solution. Each sample was placed into a spectrophotometer
calibrated at 550 nm and its absorbance was read. The following table of data was produced:
Concentration of
Potassium Permanganate (g/L)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
Abs550nm
0.140
0.257
0.377
0.498
0.628
0.72
0.85
0.96
1.1
1.2
1.4
Absorbance at 550 nm
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
0.02
0.04
0.06
0.08
0.1
0.12
Concentration of Potassium Permanganate (g/L)
Figure 1: Standard Curve of Light Absorbance by Potassium Permanganate. Samples of known
concentration were prepared and the absorbance of each sample was measured with a
spectrophotometer. A trend line of y = 11.965x + 0.0174 with R2=0.9991 is superimposed.
Notice that the trend line shows the central tendency of the data. It does not connect the points and
does not extend beyond the range of x axis values.
How to use the curve: If a solution of unknown concentration has an absorbance of 0.358, we could
find its concentration from the standard curve. Its concentration would be about 0.028 g/L.
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Calibrating and Reading the Spectrophotometer
All instruments must be calibrated before use so that they will register the same
type of information each time we make a measurement. Spectrophotometers are
usually calibrated against the solvent of the system being tested as follows:
1. Set the spectrophotometer to the correct wavelength.
2. When there is no cuvette in the machine, the B&L spectrophotometers
insert a solid block between the light source and the photocell so that no
light can pass through. Therefore, rotate the zero set knob (the one on
the left) to set the spectrophotometer to zero transmittance (infinite
absorbance) with no cuvette in the machine.
3. Fill a cuvette (small, thin walled, glass tube) with at least 5 ml of the
solvent being used. For our first experiment, the solvent is tap water.
Be sure there are no air bubbles or floating objects in the solution, and
wipe all drops and finger prints off the outside of the cuvette so that the
light will only pass through glass and solvent. Repeat the steps of
removing bubbles and fingerprints every time you use a cuvette.
4. Handling the cuvette only by the upper rim, place it into the
spectrophotometer.
5. Set 100% transmittance (zero absorbance – right hand knob) when the
cuvette containing only solvent is in the machine. This should define the
conditions under which the most light possible will pass through
solutions during the experiment (since experimental samples should
contain both solvent and solute).
6. Place 5 ml of an experimental sample into a cuvette. Remove any
bubbles, drops, or fingerprints from it and place it into the machine with
the brand mark facing you. Read the absorbance of the sample to
three significant digits and record the numbers in a table in your
notebook.
Notice that absorbance has a logarithmic scale and is read from right to left. The
scale can be read to three significant digits from 0 to about .6 absorbance units.
From .7 to 1.0 abs units, only two digits can be obtained, and data above 1.0 abs
units are not very precise.
Absorbance (Abs.) is the name of a variable and of its units. There is no further
unit name for absorbance.
Practice using the spectrophotometer until its operation is familiar and easy for
you.
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DILUTION OF METHYLENE BLUE SOLUTIONS AND ABSORBANCE
MEASUREMENTS FOR STANDARD CURVE PREPARATION
Each group of 4-5 students should produce one set of dilutions and obtain
the absorbance readings for those samples. Group data will be collected
and combined so that variation in class data can be analyzed.
1. Obtain about 3 mL of Methylene Blue stock solution from the lab bench.
Its concentration is 0.1%. Identify the test tubes (larger), cuvettes (smaller
with much clearer glass), measuring pipets, pipetting bulb, and two
beakers on the bench in front of you. Fill the larger beaker with tap water.
2. Use the table on the next page to plan a series of dilutions of the 0.1%
Methylene Blue stock solution. Begin by preparing 30 mL of a 1:10 dilution
of the stock solution in the smaller beaker. Then use this diluted solution
to prepare 5 mL volumes of all of the less concentrated solutions in the
test tubes.
Sample Calculation: Making the 1:10 dilution of the stock solution yields
a concentration of 0.01% Methylene Blue. Let’s say that you wish to
produce a 5 mL solution that is 0.006% Methylene Blue. You can use the
following equation to calculate the amount of the 0.01% solution to use:
Concentration1 x Volume1 = Concentration2 x Volume2
Let the 0.01% solution be Solution 1 and the 0.006% solution be solution
2. Then the equation will read 0.01% x Volume1= 0.006% x 5 mL, and
Volume1= 3 mL. This means you would use 3 mL of the 0.01% solution
and add 2 mL of water to produce a 5 mL solution of 0.006% Methylene
Blue.
3. Prepare the dilutions in the series, one at a time, by pipetting measured
amounts of methylene blue solution and tap water into test tubes (not
cuvettes). Use pipetting bulbs; do not pipet by mouth. Remember to
calculate all dilutions so as to produce concentrations with ONE significant
figure, since your starting concentration has just one significant digit.
Record all of your activities, both in dilution preparation and reading
absorbance values, in the table on the next page.
4. Mix the contents of the large test tube by pouring it into a cuvette. Be
certain that you have at least five mL of solution so that the solution
completely fills the light path in the spectrophotometer chamber.
5. After calibrating the spectrophotometer with tap water (repeat every
twenty minutes or so), and removing any bubbles, drops, or fingerprints,
place the cuvette into the spectrophotometer chamber. Measure and
record the absorbance of each solution at 550nm. Remember to watch for
the correct number of significant digits.
6. Repeat the above until you have obtained at all 10 data points with
concentrations from 0.001% to 0.01% as shown on Table 1. Ideally, the
absorbance values should fall in the range of ~0.05 to >1.0.For inclusion
in a spreadsheet of class data, report the last two columns of numbers
from your table to your instructor, that is, report “final concentration” and
“absorbance”.
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Table 1. Data table for recording all steps in the procedures used to prepare
samples of Methylene Blue of known concentration and to measure their
absorbance at 550nm. The last two columns of data will be used for the
construction of the Standard Curve. Report this data to your instructor at the end
of the laboratory period.
Dilution Procedure Used
Starting
concentration
of Methylene
Blue (%)
Amount of
Methylene
Blue
Solution
Amount of
Tap
Water
Added
Total
Volume
0.1
(stock solution)
3 mL
27 mL
30 mL
Final
concentration
of Methylene
Blue (%)
0.01
(Read Abs first,
then use for all
other dilutions)
0.01
5 mL
0.009
0.01
5 mL
0.008
0.01
5 mL
0.007
0.01
5 mL
0.006
0.01
5 mL
0.005
0.01
5 mL
0.004
0.01
5 mL
0.003
0.01
5 mL
0.002
0.01
5 mL
0.001
10
Abs 550nm
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C. DNA QUANTITATION
Each student, still working in a group of 4 or 5, will determine the
absorbance of the strawberry DNA solution produced in Part A.
1. Follow the procedures in Part B for calibrating the spectrophotometer at
550 nm with tap water as the solvent.
2. Removing any bubbles, drops, or fingerprints, read the absorbance of
each student’s blue-stained strawberry DNA solution at 550 nm. Record
the data in Column 2 of Table 2, below.
3. (After the laboratory period.) Complete Table 2 by following the directions
and applying the conversion factors described beneath it.
4. Clean your working space by rinsing all glassware and leaving it inverted
to dry. Wipe any spilled liquids from the bench. Discard all solids in the
trash. Pour any blue-stained solutions in the chemical waste bucket. Turn
off the spectrophotometer, cover it, and coil its cord.
Table 2: Data table for determining the amount of DNA extracted from
strawberries. Each row represents a sample prepared by a different student.
1
Student
Name
2
Abs550 for
Methylene
Blue-stained
DNA solution
3
Concentration
of Solution in
%
(from standard
curve)
4
Concentration
of DNA in
micrograms/mL
5
Total Amount
of DNA
Recovered
(micrograms)
Column 3: Determine the concentration of the solution by using the standard
curve values generated with the data from Table 1. This value is expressed as a
percentage of methylene blue, corresponding to the x-axis of the standard curve.
Column 4: For this calculation you need a conversion factor that relates the
concentration of DNA in solution to the percentage of methylene blue detected by
the standard curve assay. Use the following relationship: 0.001% Methylene Blue
corresponds to 0.086 micrograms/milliliter of DNA. This value was determined by
comparing methylene blue staining with an alternate method of measuring DNA
concentration.
Column 5: The value in Column 4, in micrograms per mL, should be corrected for
the total volume of 5 mL for the blue-stained solution.
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Writing Laboratory Report 1:
Standard Curve Analysis for DNA Quantitation
Laboratory Report 1 will emphasize the presentation and interpretation of data
collected from the strawberry DNA extraction and standard curve analysis. There
will therefore be three sections, results, discussion and conclusion. Refer to the
lab manual section on “How to Write a Scientific Report” for general information
on what to include in these sections. “Introduction to Error Analysis”, provided at
the end of this document may also be helpful. Specific details relating to this
report are provided below.
For this report, the Results section will contain the following:
1. Text: Present a paragraph that introduces the reader to the way in which the data
will be presented and describes what the reader should notice about each figure
and table. You can comment on the mathematical relationships between the
variables that are shown in the figures. Provide objective information, without any
interpretation in this section. Note, the results text for each figure is separate
from the caption of the figure.
2. Figure 1: Using your group’s data, prepare a graph of Absorbance versus
Concentration for the data compiled in Table 1 of the lab manual instructions.
Include only the data for your lab group. Be sure that the independent variable
(concentration of Methylene Blue) is on the X-axis and the dependent variable
(Abs550nm) is on the Y-axis. Use a graphing program like Excel to plot the graph
and insert a trend line. Write a full caption, including a title and brief description of
the data presented, and place it below the figure. It should describe the central
concept of the graph and show the source and purpose of the data. Be sure to
include the equation of the trend line in the caption.
3. Figure 2: Using data from the class spreadsheet, prepare a graph of the
AVERAGE (mean) Absorbance versus Concentration for the standard curve
analysis. Use Excel to calculate mean values for absorbencies at each
concentration as well as the standard deviation for these values. Each mean
value will be a point on the graph. Be sure that the independent variable
(concentration of Methylene Blue) is on the X-axis and the dependent variable
(Mean Abs550nm) is on the Y-axis. The standard deviation values will be
represented by error bars on the graph. Write a full caption, including a title and
brief description of the data presented, and place it below the figure. It should
describe the central concept of the graph and show the source and purpose of
the data.
4. Table 1: Use this format to present the data related to DNA Quantitation of your
own strawberry DNA extraction. You may use the layout of Table 2 from the
lab manual instructions or apply another design of your choice. Be sure that the
format allows you to report all of the information from Columns 2-5 of Table 2.
Your table will have a single row of data, showing the yield you obtained from a
single strawberry. Make sure that you use your group’s standard curve (Figure 1
in this report) to determine the concentration in Column 3 of this table. You do
not need to present calculations, but can do so if you prefer. Write a full caption,
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including a title and brief description of the data presented, and place it above the
table.
For this report, the Discussion section should be an integrated interpretation of
the data presented. Although the specifics are provided here in outline form,
your discussion should be written as a series of paragraphs that form a cohesive
unit for describing your data analysis.
1. Figure 1: Describe the relationship between the variables that is observed
in this graph. What is the basis for this relationship? Consider events at
the molecular level that provide the mechanism for this relationship. How
well does the data fit the expected pattern? By what criteria are you
evaluating the fit of the data? What are some possible sources of error
that could have contributed to deviations from the expected values?
2. Figure 2: When interpreting this graph, focus on the variability
demonstrated by the data. How well does the data fit the expected
pattern? By what criteria are you evaluating the fit of the data? What can
be determined from Figure 2 that cannot be seen from Figure 1? What are
some possible sources of error that could have contributed to deviations
from the expected values? [Note that you do not need to repeat your
analysis of the relationship between the variables in Figure 2 unless you
wish to present some additional information that was not pertinent to
Figure 1.]
3. Table 1: Describe the usefulness of standard curve analysis in
determining the amount of DNA extracted from strawberry cells. How can
you be sure that you were measuring DNA as opposed to another cellular
component? Describe how you might try to determine whether the amount
of DNA extracted represents a reasonable yield. What are some possible
sources of error that could have occurred in this portion of the
experiment?
The Conclusion section should be a concise summary of the relationship(s)
observed between the variables studied. Be sure that each relationship is
precisely described, using mathematical terms when possible. A comment
regarding the variation observed for the class data would also appropriately
reflect the goals of this lab exercise.
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Appendix: Introduction to Error Analysis
Working scientists soon discover that many measurements do not exactly
represent the object or process being measured. It requires skill, some insight,
and quite a bit of creativity to understand where the additional information may
have come from, how it influenced the measurement, and whether a
representative measurement can be derived from the flawed one. This task
makes up a very large part of the analysis of any experiment so understanding
and practicing it is a very important skill to develop. Statistics are used to provide
consistent methods for treating data and, since these methods are widely used,
the characteristics of data treated by these methods are well understood.
MEASUREMENTS AND ERROR
In biology, as in all sciences, measurements play a key role in the experimental
process. A measurement is a precise and objective description or representation
of a real phenomenon. In order for measurements to be meaningful, they must
be reliable. Reliability is established by defining the conditions under which
measurements are taken and examining the variability in the resulting
measurements. Usually measurements are repeated enough times to insure
confidence that reproducible values can be obtained. Examination of the
variability also involves a calculation of the limits of accuracy in the measuring
procedure. In this experiment, we will measure the absorbance of solutions of
known concentrations of Methylene Blue and will use the calculated value of one
standard deviation as the limit of accuracy among the measurements.
Error can be conceived of as variation from an unknown and immeasurable,
hypothetically true value that is usually taken to be the mean or the most frequent
measurement in a set of data. Error is a part of any measurement and variation is
a property of any set of data. Its sources are usually intrinsic (inherent variability
in the subject being measured, which is sometimes quite large in biological
studies) and/or systematic (variability due to the measurement process).
One of the great achievements of modern science has been to show that
variations (i.e. errors) in measurements are strikingly similar regardless of the
object being measured. This general similarity has been incorporated into the
Normal Law of Error, which allows us to estimate the measurement error itself
and to analyze the nature of variation. The task now is to use this law to describe
in precise mathematical terms the distribution of a specific measurement taken in
the class in such a way that the common features of the sample group and the
variability are revealed. How does the frequency distribution of this
measurement compare to the usual bell-shaped normal curve of error? One
possible distribution is shown in Figure 1 below.
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100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
3s
2s
1s
X
1s
2s
3s
Figure 1: Example of a normal curve of error. This
hypothetical curve was constructed by assigning an
arbitrary value to the mean and a percentage of that value
to the appropriate standard deviation multiples away from
the mean.
As Figure 1 suggests, most sets of measurements exhibit two identifying
characteristics: one, a tendency to group around a central value (the mean is
usually the most probable measurement), and two, a degree of variability or
spread away from the central value (the standard deviation is a measure of the
variability--or error--in the set of measurements.) The Normal Law of Error
defines these two characteristics as follows:
1. The mean - the average value obtained by dividing the sum of a sample of
quantities by the total number of quantities added. For a set of N measurements
__
with values symbolized X1, X2, X3,----XN, the mean (X) is calculated as follows:
_
X =  Xi/N
where Xi stands for the individual measured values from 1 to N.
2. The standard deviation - a measure of variability relative to the mean, whose
formula is:
________
/
_
s = /(Xi-X)2
√ N-1
_
where s is the standard deviation, Xi - X is the difference between each value
and the mean (these are squared to eliminate any negative values), and N is the
total number of individual measurements.
The mean and the standard deviation are two constants whose value can be
calculated easily from any set of data in which a single variable has been
measured many times under a single set of conditions. Together they are
15
Standard Curve Analysis for DNA Quantitation
Colavito and Logan, 2010
usually sufficient for the reconstruction of the distribution of data in the form of a
curve fitted to the histogram whose height is proportional to the mean and whose
width is determined by the standard deviation. More importantly, these two
constants serve as a shorthand representation of the original sample, thereby
furnishing a concise basis for the interpretation and comparison of
measurements.
The standard deviation is a mathematically defined characteristic that is based
on the number of measurements that should fall within any given interval
centered on the mean. This function is pictured in Figure 1, showing a normal
curve of error in which the horizontal scale is marked off in multiples of the
standard deviation on either side of the mean. In any normal distribution, the
expected distribution of measurements is approximately:
66% of the values lie within mean ± ls
95% of the values lie within mean ± 2s
99% of the values lie within mean ± 3s
The method of calculating the mean and standard deviation is shown in Table 1
for a small sample of five hypothetical measurements.
Table 1: Sample Calculation of Mean and Standard Deviation from a set of five
data measurements.
Measurement
Measurement
minus mean
Square of
difference
_
Xi –X
7
-1
-4
3
-5
_
(Xi – X)2
49
1
16
9
25
100
Xi
35
27
24
31
23
140
_
Mean = X = 140 = 28
5
__
Standard Deviation = S = / 100 = √25 = 5
√ 5-1
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Standard Curve Analysis for DNA Quantitation
Colavito and Logan, 2010
At this point it is worth emphasizing the steps taken in the process of condensing
a collection of measurements into a useful form. These steps are:
1.
Sorting the raw data into arbitrary groups defined by regular intervals
2.
Reporting the grouped and ordered data in a table or histogram.
3.
Calculating the mean and the standard deviation.
4.
Analyzing the measurements to see their pattern of variability and
describing the variability characteristics in the data sample.
17