Beer-Lambert Law (Video)

0:26 Why biology labs measure light

First, we will look at why biology labs measure light. Several routine steps in a molecular biology laboratory depend on a measurement of light. Before a PCR or a sequencing run, you check the concentration of the DNA sample, and before you load a protein gel, you check the protein concentration in the same way. The density of a bacterial culture, which decides when you induce expression or harvest the cells, also comes from a light reading on the same instrument. Each of these measurements passes light through the sample and records how much light arrives at a detector. The Beer-Lambert law converts that light reading into a concentration, and the conversion needs the absorption strength of the substance and the distance the light travels through the sample.

1:28 Light, wavelength and the spectrophotometer

Second, we will cover light, wavelength and the spectrophotometer. Because the law describes what the instrument records, the explanation turns first to light and to the instrument itself. Wavelength is the distance between successive peaks of a light wave, and you give wavelength in nanometers. A spectrophotometer passes light of a chosen wavelength through a sample and measures how much of that light reaches a detector. Most biological measurements use the ultraviolet range, from about 200 to 400 nanometers, or the visible range above 400 nanometers. A molecule absorbs light at wavelengths that match the energy gaps between its electron orbitals, so the absorption spectrum characterizes the substance. The bases of nucleic acids absorb strongly near 260 nanometers. The aromatic side chains of tryptophan and tyrosine absorb near 280 nanometers. The light path runs from a lamp through a monochromator, which selects one wavelength, then through the cuvette that holds your sample, and then onto a photodetector. The detector output is a voltage proportional to the light intensity that reaches the photodetector.

2:48 Transmittance and absorbance

Third, we will define transmittance and absorbance. The detector reports the intensity of the transmitted light, and the ratio of that intensity to the incident intensity is transmittance. Transmittance is the fraction of the incident light that passes through the sample, written T equals I divided by I0, where I0 is the intensity entering the sample and I is the intensity leaving the sample. Absorbance is minus the base-10 logarithm of transmittance, written A equals minus log10 of T, and absorbance carries no units. If half of the light passes, T is 0.5 and A is 0.30, and if one tenth passes, T is 0.10 and A is 1.0. When only one hundredth of the light passes, T falls to 0.01 and A rises to 2.0. Each additional unit of absorbance therefore cuts the transmitted light by a further factor of ten. The logarithm appears because each thin layer of the sample removes the same fraction of the light that reaches that layer. Transmittance therefore falls exponentially as the sample gets thicker, and the logarithm of transmittance falls in a straight line.

4:03 Where the law comes from

The fourth part traces where the law comes from. The law combines two observations that were made separately. In the eighteenth century, Pierre Bouguer and then Johann Heinrich Lambert showed that absorbance rises in proportion to the distance the light travels through the sample. In 1852, August Beer showed that absorbance rises in proportion to concentration in the same way. The combined relationship carries the names of Lambert and Beer, and the equation you use on the bench contains both proportionalities.

4:49 The equation A = epsilon l c

Fifth, we will build the equation A equals epsilon l c. The two proportionalities join into one equation, A equals epsilon times l times c. The symbol epsilon stands for the molar absorption coefficient, which states how strongly a substance absorbs at one wavelength, in units of per molar per centimeter. The symbol l stands for the path length, the distance the light travels through the sample, which is usually 1 centimeter in a standard cuvette. The symbol c stands for the concentration of the absorbing substance, in moles per liter, which is the same as molar. The units of the product are per molar per centimeter times centimeter times molar, and those units cancel to leave a dimensionless number, which matches the definition of absorbance as a logarithm. Rearranged for concentration, the equation reads c equals A divided by the product of epsilon and l. You measure A on the instrument, you look up epsilon for the substance at that wavelength, and you know l from the cuvette.

5:54 The spectrum and the choice of wavelength

Sixth, we will look at the spectrum and the choice of wavelength. The molar absorption coefficient belongs to one wavelength, so the choice of wavelength comes before any calculation. An absorbance spectrum plots absorbance against wavelength for one sample, and the highest point of the curve marks the wavelength of maximum absorbance. You usually measure at that maximum, because the sample gives its largest signal there and small errors in the wavelength setting change the reading least where the curve is flat. On a steep side of the peak, a shift of 1 to 2 nanometers changes epsilon noticeably, and the calculated concentration changes with epsilon. For DNA the maximum lies near 260 nanometers. NADH, the coenzyme in the worked example that follows, absorbs most strongly near 340 nanometers, outside the region where proteins and nucleic acids absorb strongly, which is why enzyme assays follow NADH at that wavelength. A coefficient taken from a reference table therefore applies only at the wavelength stated with it, in the same solvent and at the same pH.

7:07 Path length and concentration

Seventh, we will see how path length and concentration change absorbance. The equation predicts how absorbance responds to each of the two variables you control. Two identical sunglass lenses stacked together darken the view as much as one lens of twice the thickness, because each lens removes the same fraction of the light that reaches that lens. A single lens with a higher concentration of dye darkens the view in the same way as a thicker lens. At a fixed concentration, doubling the path from 1 centimeter to 2 centimeters doubles the absorbance from 0.30 to 0.60. Transmittance falls from 50 percent to 25 percent over the same change, because 0.5 multiplied by 0.5 gives 0.25. Absorbance values add across the two path lengths, and transmittance values multiply. Doubling the concentration at a fixed path length has the same effect on absorbance as doubling the path length.

8:12 Worked example with NADH

The eighth part works through an example with NADH. To apply the rearranged equation, consider an enzyme assay with illustrative numbers. NADH absorbs at 340 nanometers, and its molar absorption coefficient at that wavelength is 6,220 per molar per centimeter. You read the absorbance of the sample at 340 nanometers in a 1 centimeter cuvette and obtain 0.311. The rearranged equation from the previous segment applies directly. You substitute 0.311 for A, 6,220 per molar per centimeter for epsilon, and 1 centimeter for l. The product of epsilon and l is 6,220 per molar. You divide 0.311 by 6,220 per molar and obtain five point zero times ten to the minus five molar. Five point zero times ten to the minus five molar equals 50 micromolar.

9:15 The linear range and deviations

Ninth, we will cover the linear range and deviations from the law. The equation describes a straight line only within limits. The law holds best for dilute solutions. Above an absorbance of about 1 to 1.5 on many instruments, stray light and the small fraction of light that reaches the detector bend the plot of absorbance against concentration downward. Particles and bubbles scatter light out of the beam and add to the reading. A substance that associates into dimers at high concentration has a different coefficient from the free molecule, so absorbance stops tracking concentration. Light that contains more than one wavelength also bends the plot, because epsilon differs across the band. When a reading falls above the linear range, you dilute the sample into the range, read the absorbance again, and multiply the calculated concentration by the dilution factor.

10:14 Measuring DNA and RNA at 260 nm

Tenth, we will measure DNA and RNA at 260 nanometers. The same equation applies at 260 nanometers, a wavelength used routinely to measure DNA and RNA. For double-stranded DNA, an A two sixty of 1.0 in a 1 centimeter path corresponds to about 50 micrograms per milliliter. For RNA the figure is about 40 micrograms per milliliter, and for single-stranded DNA about 33 micrograms per milliliter. In an illustrative example, a DNA sample diluted 1 in 50 reads an A two sixty of 0.45. You multiply 0.45 by 50 micrograms per milliliter to get the concentration in the diluted sample, which is 22.5 micrograms per milliliter. You then multiply by the dilution factor of 50 and obtain 1,125 micrograms per milliliter in the undiluted sample. The factor of 50 micrograms per milliliter already contains epsilon and the 1 centimeter path length, so you multiply the reading by the factor instead of dividing by a coefficient. The factor applies only at 1 centimeter, and a different path length changes the number.

11:31 Purity ratios

The eleventh part covers purity ratios. The same instrument also shows what else is present in the tube. You read the absorbance at 260 nanometers and at 280 nanometers on the same sample and divide the first reading by the second. A sample with an A two sixty of 0.90 and an A two eighty of 0.50 gives a ratio of 1.80, which matches the value for pure DNA. Pure RNA gives a ratio of about 2.0. A ratio well below 1.8 suggests contamination with protein or with phenol. You also divide the A two sixty by the absorbance at 230 nanometers. Clean nucleic acid gives an A two sixty over A two thirty ratio of about 2.0 to 2.2, and lower values suggest carry-over of guanidine salts or carbohydrate. A ratio tells you about contamination, and it does not correct the concentration you calculated from the A two sixty.

12:35 Measuring protein at 280 nm

Twelfth, we will measure protein at 280 nanometers. Proteins absorb in the ultraviolet because of their aromatic side chains. Tryptophan and tyrosine carry the absorption at 280 nanometers, so the molar absorption coefficient at 280 nanometers differs between proteins. Bovine serum albumin has a coefficient at 280 nanometers of about 43,824 per molar per centimeter and a mass of 66,430 grams per mole. A solution of 1 milligram per milliliter of bovine serum albumin therefore reads an A two eighty of about 0.66 in a 1 centimeter cuvette. A sample of the same protein reading 0.33 holds about 0.5 milligrams per milliliter.

13:29 A protein coefficient from its sequence

Thirteenth, we will estimate a protein coefficient from its sequence. When no published coefficient exists for your protein, you can estimate the coefficient from the amino acid sequence. At 280 nanometers each tryptophan contributes about 5,500 per molar per centimeter, each tyrosine about 1,490, and each disulfide bond between two cysteines about 125. You add these contributions to obtain the molar absorption coefficient of the folded protein. In an illustrative protein with three tryptophans, twelve tyrosines and two disulfide bonds, the sum is 16,500 plus 17,880 plus 250, which gives 34,630 per molar per centimeter. Dividing by the molar mass of 25,000 grams per mole gives the absorbance of a 1 milligram per milliliter solution, about 1.39 in a 1 centimeter cuvette. A sample of this protein reading an A two eighty of 0.69 therefore holds about 0.5 milligrams per milliliter. Coefficients estimated from the sequence usually agree with measured values to within a few percent for proteins that contain tryptophan, and the estimate grows less reliable for proteins without tryptophan. When the sequence is unknown or the sample holds a mixture of proteins, you measure the concentration with a colorimetric assay against a standard curve.

15:07 Cell density with OD600

Fourteenth, we will read cell density with OD six hundred. A bacterial culture is read on the same instrument, though the physics differs. OD six hundred is the optical density at 600 nanometers, a measure of light scattering by a cell suspension. A cell in the beam sends light out of the forward direction, so less light reaches the detector, and the instrument reports that loss as absorbance. An OD six hundred of 1.0 corresponds to very roughly 8 times 10 to the 8 E. coli cells per milliliter on a typical instrument. The reading depends on the optics of the instrument and on the size of the cells, so a laboratory calibrates its own conversion against viable plate counts. Readings stay proportional to cell number only up to roughly 0.4 to 1.0, so you dilute a dense culture before you read the absorbance. You multiply the diluted reading by the dilution factor to report the culture density.

16:11 Blanks, cuvettes and microvolume readings

The fifteenth part covers blanks, cuvettes and microvolume readings. A correct calculation still needs a correct reading. A blank is a cuvette that holds only the buffer or solvent, and you use the blank to set the zero of absorbance on the instrument. The blank must contain the same buffer as the sample, because the buffer itself absorbs at short wavelengths. Below about 300 nanometers, ordinary plastic and glass absorb ultraviolet light, so you use a quartz cuvette or a cuvette the manufacturer marks as UV-transparent. You fill the cuvette and wipe the outside, and you place the clear faces toward the beam. A microvolume instrument places a drop of 1 to 2 microliters between two surfaces separated by a path of 1 millimeter or 0.2 millimeters. The instrument scales the reading to a 10 millimeter path before it reports the absorbance, which extends the usable range for concentrated nucleic acid.

17:13 Troubleshooting absorbance readings

The last part covers troubleshooting absorbance readings. When a reading looks wrong, the cause usually lies in the sample or in the setup. A reading above the linear range gives a concentration that is too low, so you dilute and read the sample again. A blank of the wrong buffer shifts the zero and offsets each reading. Bubbles, fingerprints and scratches on the cuvette scatter light and raise the absorbance. A turbid sample scatters light in the same way, and you centrifuge the sample or filter the sample before you read the absorbance. A plastic cuvette at 260 nanometers absorbs the light you are trying to measure. A sample that you did not mix gives a reading that varies between repeats. A wrong coefficient or a forgotten dilution factor in the calculation gives a concentration that is wrong by a fixed factor.

18:09 Recap and quiz

Before the question, here is what this video covered. Absorbance is minus the base-10 logarithm of transmittance, and the Beer-Lambert law sets absorbance equal to epsilon times l times c. Path length and concentration each raise absorbance in proportion, and the linear range limits where that proportionality holds. At 260 nanometers, an A two sixty of 1.0 in a 1 centimeter cuvette corresponds to about 50 micrograms per milliliter of double-stranded DNA. The purity ratios A two sixty over A two eighty and A two sixty over A two thirty report contamination in the same tube. OD six hundred measures scattering by cells, and the conversion to cell number needs a calibration on your own instrument. Here is a quiz question for you. Consider a solution with an absorbance of 0.48 in a 1 centimeter cuvette, measured at a wavelength where the molar absorption coefficient is 12,000 per molar per centimeter. Find the concentration in micromolar. Take a moment to work it out, then compare with the answer. The concentration is the absorbance divided by epsilon times l, so 0.48 divided by 12,000 gives 4 times ten to the minus five molar, which is 40 micromolar.