How to Make a Standard Curve (Video)

0:03 Introduction

In this video, you will learn how to make a standard curve and how to use it to basically determine an unknown concentration using a standard curve. A standard curve relates the signal of samples of known concentration to their concentration, so you can read the concentration of an unknown sample from its signal.

0:25 Why assays need a standard curve

First, we will look at why assays need a standard curve and separate standards from unknowns. An assay reports a signal, such as absorbance at a set wavelength, fluorescence counts or a threshold cycle from a qPCR instrument. The signal has no fixed numerical relationship to concentration from one run to the next. Its size depends on the assay chemistry, the temperature, the incubation time and the batch of reagent, so the same concentration can give a different reading from one run to the next. A standard curve works in the same way as a ruler you make yourself, which you mark at lengths you already know before you read an unknown object against the marks. A length beyond the last mark cannot be read from that ruler, and a signal beyond the last standard cannot be converted into a concentration. Running a set of standards alongside the unknowns in the same run gives the relationship between signal and concentration for that run. The standards and the unknowns then share the plate, the reagents and the incubation. You use that relationship to convert the signal of each unknown into a concentration.

1:50 Standards and unknowns

The curve rests on a clear distinction between two kinds of sample. A standard is a sample of known concentration that you use to calibrate the assay. An unknown is a sample whose concentration you want to find. You set the concentration of each standard, so concentration is the independent variable and it goes on the x axis. You measure the signal, so the signal is the dependent variable and it goes on the y axis. In a Bradford assay the independent variable is protein concentration in micrograms per milliliter and the dependent variable is absorbance at 595 nanometers, while in qPCR the x axis carries the logarithm of the starting copy number and the y axis carries the threshold cycle. The axes change with the assay, and the logic of the plot stays the same.

2:42 Preparing the standards

Second, we will prepare the standards. The quality of the curve depends on the quality of the standards. You prepare the standards from a stock solution of known concentration by serial dilution, the procedure covered in the serial dilution video. For the Bradford example in this lesson, the stock holds 2 milligrams per milliliter of bovine serum albumin, and the standards sit at 0, 125, 250, 500, 750, 1,000 and 1,500 micrograms per milliliter. The zero standard is a blank, and it contains the buffer and the reagent without any protein. You run each standard in more than one well, so the replicates show the pipetting error at that level. You prepare the standards in the same buffer as the samples, because a different salt concentration or a different amount of detergent changes the reading. You space the standards so that no gap lies in the part of the range where you expect the unknowns. A gap at the top of the range leaves the high samples poorly calibrated. You change tips between dilution steps, and you mix each tube before the next transfer. The standards go through the same pipetting steps, the same incubation and the same plate reader as the samples. You write the nominal concentration of each standard on the plate map, together with its dilution factor.

4:14 The straight line y = mx + b

Third, we will cover the straight line, y equals m x plus b, then fit the line and read R squared. Because the standards define the relationship, the next step concerns its form, and many assays respond in proportion to concentration over a working range, so a straight line describes the standards across that range. You write the line as y equals m x plus b. The slope, m, gives the change in signal for each unit change in concentration, and its units are absorbance units per microgram per milliliter in the Bradford example. The intercept, b, gives the signal the line predicts at zero concentration, and it sits close to the reading of the blank. A positive intercept means the reagent alone produces some signal. The slope carries the sensitivity of the assay, and a steeper slope means a larger change in signal for the same change in concentration. The line lets you move in both directions. You put a concentration into the equation and get the signal the assay should produce, and you put a measured signal into the rearranged equation and get a concentration. The slope and the intercept belong to one run on one plate with one batch of reagent.

5:31 Fitting the line and reading R squared

Real points never sit exactly on a line, so you fit the line that passes closest to the set of standards. The least-squares method chooses the slope and the intercept that minimize the sum of the squared vertical distances from each point to the line. The vertical distance from a point to the line is its residual. Squaring the residuals stops a point above the line from canceling a point below the line, and it gives more weight to the points that sit far from the fit. R squared, the coefficient of determination, is the fraction of the variation in the signal that the fitted line explains. A value of 1 means the line explains all of the variation, and a value of 0 means the line explains none. A high R squared does not prove that the relationship is straight. A gently curved set of points can still give an R squared of zero point nine nine. You look at the residuals as well, plotted against concentration. In a good fit the residuals scatter above and below zero with no pattern. Residuals that are positive at both ends of the range and negative in the middle, or the reverse, show curvature, and a straight line misreads both ends of that curve. Spreadsheet and plate-reader software returns the slope, the intercept and R squared from the standard wells.

6:54 Worked example with a Bradford assay

Fourth, we will work through an example with a Bradford assay. The worked example uses a Bradford assay with illustrative numbers. The standards at 0, 125, 250, 500, 750, 1,000 and 1,500 micrograms per milliliter give absorbances at 595 nanometers of zero point zero one two, zero point zero eight three, zero point one five eight, zero point three zero six, zero point four six two, zero point six one one and zero point nine zero three. A least-squares fit through those seven points gives a slope of zero point zero zero zero five nine seven absorbance units per microgram per milliliter and an intercept of zero point zero one zero. R squared is zero point nine nine nine nine. The blank absorbance of zero point zero one two sits close to the fitted intercept of zero point zero one zero. An unknown lysate diluted one in five reads zero point six one eight, zero point six two four and zero point six one nine in triplicate. The mean of the three readings is zero point six two zero, and the coefficient of variation across the triplicate is zero point five percent. The mean absorbance falls between the readings of the 1,000 and 1,500 standards, so the sample sits inside the calibrated range. You rearrange the equation to solve for concentration, which gives x equals y minus b, divided by m. Substituting the mean absorbance and the fitted values gives x equals zero point six two zero minus zero point zero one zero, divided by zero point zero zero zero five nine seven. The result is about one thousand and twenty micrograms per milliliter in the diluted sample. You multiply by the dilution factor of five, and the original lysate holds about five thousand one hundred micrograms per milliliter, or five point one milligrams per milliliter. Reading an unknown from the curve inside the range of the standards is interpolation.

9:04 Staying inside the range

Fifth, we will stay inside the range of the standards and cover replicates and blank subtraction. The equation returns a number for any signal you put into it, and only some of those numbers are valid. The curve covers 125 to 1,500 micrograms per milliliter in the Bradford example. A reading above zero point nine zero three lies beyond the top standard, and the line there is an extrapolation. The assay may have saturated at that absorbance, so the true concentration is higher than the equation reports. A reading below the lowest standard is also an extrapolation, and the blank and the detection limit dominate the signal at that end. You bring the sample into the range by diluting it or by concentrating it, and then you assay it again. You dilute with the same buffer you used for the standards, and you record the dilution factor. A sample you dilute and read at the middle of the curve gives a more reliable result than the same sample read beyond the top standard.

10:11 Replicates and blank subtraction

Each reading carries random error from pipetting, from the plate and from the detector, so you repeat wells and average the readings. The mean of the replicates is the value you put into the equation. The coefficient of variation across the replicates measures their spread as a percentage of the mean, and the coefficient of variation video covers the calculation. A triplicate with a coefficient of variation near zero point five percent is tight, and a value above about ten percent points to a pipetting problem in that well. You can correct for background absorbance before you fit the line or within the fit itself. You subtract the mean blank absorbance from each standard and each unknown before you fit the line, or you keep the blank in the fit and let the intercept absorb the background. You apply the same choice to the standards and to the unknowns, because subtracting a blank from one set and not the other shifts the result. The intercept of zero point zero one zero in the worked example carries the reagent background when you fit with the blank included.

11:22 Checking the curve by back-calculation

Sixth, we will check the curve by back-calculation. A high R squared does not show that the curve reads correctly at each level, so you check the standards by back-calculation. You put each standard's absorbance back through the fitted equation and compare the result with the concentration you prepared. In the worked example the six non-zero standards return 121.8, 247.4, 495.3, 756.5, 1,006 and 1,495 micrograms per milliliter. Those values lie between 97.5 and 100.9 percent of the nominal concentrations. Many validation guidelines accept a standard within 15 percent of its nominal value, and within 20 percent at the lowest concentration on the curve. A standard that falls outside those limits points to a pipetting error or a dilution error at that level. You prepare that standard again and repeat the assay. A pattern across the standards carries more information than a single failure. Back-calculated values that drift above or below nominal in a steady direction across the range show curvature, and values that scatter without a pattern show random error.

12:46 Curved standards and the four-parameter logistic

Seventh, we will look at curved standards with the four-parameter logistic model and at the qPCR standard curve. Some assays do not respond in a straight line at all. ELISA standards form an S-shaped curve when plotted against the logarithm of concentration, flat at low concentration, steep in the middle and flat again at high concentration. You fit a four-parameter logistic model to these points. The four parameters are the bottom asymptote, the top asymptote, the midpoint and a slope factor. The midpoint is the concentration at which the signal lies halfway between the bottom and top asymptotes, and the symbol EC50 refers to that concentration. A straight line through S-shaped data misreads the flat sections at both ends, because the line keeps rising where the assay has leveled off. You read the unknowns from the steep middle section, where a small change in concentration produces a measurable change in signal. A sample that reads on the flat top section needs dilution before you can quantify the analyte. The four-parameter fit is nonlinear, so the software searches for the four values that minimize the residuals, and it needs starting estimates. You check the fit by back-calculating the standards, as with a straight line.

14:08 The qPCR standard curve

The same calibration logic applies to the quantification of nucleic acids by qPCR, with different axes. The threshold cycle, or Ct, is the cycle at which the fluorescence crosses a threshold that you set inside the exponential phase of the reaction. A sample with more template reaches that threshold in fewer cycles, so Ct falls as the starting copy number rises. You prepare a tenfold dilution series of a standard and plot Ct against the logarithm to base ten of the starting copy number. The line slopes downward, and its slope gives the efficiency of the reaction. The efficiency equals ten raised to minus one divided by the slope, with one subtracted from the result. A slope of minus three point three two gives an efficiency of 100 percent, and the Ct falls by three point three two cycles for each tenfold step. A slope of minus three point six gives an efficiency of about 90 percent. You aim for an efficiency between 90 and 110 percent. You read an unknown's copy number from its Ct through the fitted line, and you report the result with the efficiency and the R squared of the standard curve.

15:23 Troubleshooting standard curves

The last part covers troubleshooting standard curves. When the curve or the results look wrong, the cause usually lies in a few places. Extrapolating beyond the highest standard or below the lowest standard gives a number that no standard calibrates. A low R squared points to pipetting error, to a standard prepared at the wrong concentration, or to a bubble in one well. A curve that bends at the top end shows saturation of the detection chemistry, and you narrow the range or fit a nonlinear model. Standards prepared in a different buffer from the samples respond differently, and the curve then misreads the unknowns. Standards stored for too long, or diluted from a stock whose concentration was wrong, shift the whole curve. A forgotten dilution factor gives a result that is wrong by a constant multiple, and the error survives a good R squared. You check the stock concentration against the certificate of analysis before you dilute. You run a new curve with each plate, because the slope and the intercept belong to that set of reagents and that day. You record the equation, the R squared and the back-calculated standards with the results.

16:40 Recap and quiz

Here is a short review of the main points. A standard curve converts a signal into a concentration by fitting a line to standards of known concentration. You prepare the standards by serial dilution in the sample buffer, with a blank and replicates, and you fit the line by least squares. You read an unknown inside the range of the standards, and you multiply by its dilution factor. Back-calculation checks each standard against its nominal value, and a four-parameter logistic model handles S-shaped ELISA data. Test what you learned with this question. Consider a standard curve with the equation y equals zero point zero zero zero six x plus zero point zero one zero, where y is absorbance and x is protein concentration in micrograms per milliliter. A sample reads an absorbance of zero point three one after a dilution of 1 in 10. Find the protein concentration of the original sample. Pause the video and work it out. The answer follows. Subtracting zero point zero one zero leaves zero point three, and dividing by zero point zero zero zero six gives 500 micrograms per milliliter in the diluted sample, so the original sample holds 5,000 micrograms per milliliter, or 5 milligrams per milliliter.