# How to do lab calculations

You'll need to do a lot of calculations in Bio 6B. Almost all of them are either conversions or dilutions, so two basic kinds of equations will solve all your problems.

## Units of measurement

If you work in a lab, you need to pay attention to units of measurement. For the labs in this course, there are only a few kinds of units, and the prefixes that go with them.

### Fundamental units:

I'm sure you are familiar with these from your chemistry and physics classes:

• Distance: meters (m)
• Mass: grams (g)
• Volume: liters (l)

### Concentrations:

Concentration, or the amount of stuff in a given volume, can be given several ways:

• Molar (M), defined as moles of solute per liter of solution. In this lab, we’ll usually express concentrations in μM or mM.
• Percent (%) usually means mass/volume: For example, if you want to make a gel with an agarose concentration of 0.5%, you would use 0.5 g agarose per 100 ml of solution.
• g/l, or μg/μl, or similar units (mass over volume). We’ll use these units for DNA and proteins, which are variable in size.
• X. You'll use the mysterious X frequently in lab. For example, if you start with an unknown concentration of a bacterial culture, you could call it X. If you do a 10-fold dilution, you'll have a concentration of 0.1X.

### Prefixes and powers of 10

The prefixes we normally use in this course are in multiples of 1000:

• 1 mg = 1 milligram = 10-3 g
• 1 μg = 1 microgram = 10-3 mg = 10-6 g (that’s a greek letter μ, not a u)
• 1 ng = 1 nanogram = 10-3 μg = 10-9 g
• 1 pg = 1 picogram = 10-3 ng = 10-12 g

You could memorize this list, and you'll be ready to do the various conversions in the Bio 6B lab. Alternatively, you could remember: m > μ > n > p. For each step, you multiply or divide by 1000. You don’t need to remember that a nanogram is 10-9 grams; just remember that there are 1000 ng in 1 μg and 1000 μg in 1 mg and 1000 mg in 1 g.

### Significant figures

Be realistic about your significant figures; the level of precision should reflect the way you would do the experiment. Most of the lab experiments are done with a Pipetman, which has a practical precision of three significant figures. If you're using a P-2 to pipet 1 μl, you'd set it at 1.00 μl, so 1.00 μl would be the correct answer for a quiz. If the answer is 0.10 μl, you don't need to write 0.100 μl, because you don't set the pipet that way.

Also, if the correct answer is 1.00 μl, someone might give an answer like 0.001 ml or even 10-6 ml, but I wouldn't consider these answers correct because they don't reflect the way you would perform the experiment and don't have the correct number of significant figures. An answer like 0.00113045 ml would definitely be wrong; it might be mathematically correct, but it's unrealistic in its precision and impractical in terms of how you'd pipet it.

## Conversions using conversion factors

A conversion factor is the relationship between one unit and another; for example, 1000 μg = 1 ml.

Suppose you have a solution of DNA at a concentration of 1 mg/ml and you want to express that concentration in μg/ml. You'll need to use the appropriate conversion factor. The equation is:

(1 mg/ml)(1000 μg/1mg) = 1000 μg/ml

It's helpful to write it like this:

Write it all on paper, including the units. The unwanted units should cancel out, leaving you with the units you want. Many students want to reach for the calculator first, without writing out the equation; often, this leads to the wrong answer.

## Dilutions

The basic equation for figuring out how to make dilutions is: C1V1 = C2V2.

• C1 = the initial concentration of your solution.
• V1 = the amount of stock solution you need to add. Usually this is what you are solving for.
• C2 = the final concentration (usually lower than C1).
• V2 = the final volume – includes the volume in V1 plus the amount of water or buffer you add.

Before you use it, take a moment to think about what the equation says: if you start with a small amount of a solution and you add enough water to double the volume, then the concentration will be cut in half.

For example, suppose you have a stock solution of DNA at 5 mg/ml. You want to make 1 ml of DNA solution with a final concentration of 1 mg/ml. The equation would be:

(5 mg/ml)(V1)=(1 mg/ml)(1 ml)

Solve for V1; the answer is 0.2 ml. Always write down the units when you do these calculations. In this example, the units are the same on both sides of the equation, but that won't always be the case.

If you're making this solution in the lab, you would pipet 0.2 ml (200 μl) of your starting DNA solution into a micro tube, then add 0.8 ml of water or another solution to bring the total volume up to 1 ml.

In chemistry class, you might have learned this as M1V1=M2V2. This would make sense if your concentrations are given in molar (M). In the bio lab, you will often have other concentration units, so it's best to think of the equation as C1V1=C2V2 instead of M1V1=M2V2.

### Serial dilutions

Suppose you have 1 ml of a bacterial culture. The concentration of the bacteria is unknown, so you’ll have to call it 1x. You need to dilute this culture a lot, to get 1 ml of 10-6x solution. How do you do the dilution? You could try the dilution this way:

(1x)(10-6 ml) = (10-6x)(1 ml)

Mathematically, this is correct; in the lab, it's not correct. The problem is that you can't set your pipetman to 10-6 ml, or 0.001 μl (1 nl). Alternatively, you could look at it like this:

(1x)(1 ml) = (10-6x)(106 ml)

Again, this is mathematically correct, but wrong in the lab. We don't have a sterile culture tube that holds 106 ml, or 1000 liters. The only practical way to do large dilutions is to do a series of dilutions: make a 10-2x dilution, use that to make a 10-4x dilution, then use the 10-4x dilution to make the 10-6x. For example:

(1x)(10 μl) = (10-2x)(1000 μl)

(10-2x)(10 μl) = (10-4x)(1000 μl)

(10-4x)(10 μl) = (10-6x)(1000 μl)

You would need to do three dilutions consecutively, each time using the diluted culture to make a further dilution. In this example, I put the volumes in microliters, because that's how our micropipets work.

## Adding ampicillin: the milli x micro = micro x milli trick

Suppose you want to add some ampicillin (an antibiotic) to a bacterial culture. You need a low final concentration of amp in your culture (100 μg/ml), and it comes as a highly concentrated solution (100 mg/ml). As an example, your calculation might look like this:

(100 mg/ml)(V1) = (100 μg/ml)(4 ml)

You want to solve for V1, the initial volume; this is the amount you'll add to the 4 ml of LB broth that's already in the tube. There's a slight problem here; the concentration of the stock solution (C1) is in mg/ml and the final concentration is in μg/ml. Also, what units should you use for V1? If you use ml, it matches V2 but you'll need to convert it to μl because that's how our micropipettes work.You could do a conversion like this:

(100,000 μg/ml)(V1) = (100 μg/ml)(4 ml)

Solve for V1 and you get 0.004 ml. Convert that to μl, and it's 4 μl. That works, but there is a simpler way. You could do it like this instead:

(100 mg/ml)(V1 μl) = (100 μg/ml)(4 ml)

The answer is V1=4 μl. The unit prefixes don't match, but they don't need to. On the left side of the equation, you have milli x micro; on the right side of the equation, you have micro x milli. It comes out the same. Since the prefixes represent powers of 10, you could replace the prefixes with their powers of 10:

(100 x 10-3 g/ml)(V1 x 10-6 l) = (100 x 10-6 g/ml)(4 x 10-3 l)

Again, you don't need to do any conversions here; 10-3 x 10-6 = 10-6 x 10-3. This little trick only works in specific cases, but those cases turn up often in the lab.

When you visualize protein on an SDS-PAGE gel, you need to load the appropriate mass of protein (micrograms) in order to make your gel turn out right. Since you're measuring out the protein sample with a micropipet (in microliters), you'll need to convert from mass to volume, using the protein concentration. For an explanation of how to do this, see calculations for protein gel loading.

## Calculating how much DNA to use

Suppose you’re doing a restriction digest. You know you want 1 μg of DNA to complete the next phase of your experiment, but your DNA is in solution and you're going to pipet it. How many microliters equals 1μg? You'll need to use a conversion factor, which is the concentration of your DNA solution. Here's how to solve it.

Write down what you know:

1 μg = ? μl

You're given an amount in micrograms and you need to convert that to microliters. To do the conversion, you need to know the relationship between micrograms and microliters; this is the concentration of your starting DNA solution in μg/μl. Before you think about the numbers, write down the equation with the appropriate units:

From this, it's obvious that you need to invert the concentration in order to make the units cancel out. Failing to invert the concentration is by far the most common mistake on this type of problem. Suppose the concentration is 0.5 μg/μl. Write the equation like this:

It's a simple calculation, but it's worth writing it down with the units before you reach for the calculator. In fact, if you take a moment to think about this particular example, I hope you'll find that you don't need a calculator to solve it. A calculator is a wonderul tool for helping you to avoid thinking; however, in science it often turns out that thinking is a good thing.

## Calculating how much enzyme to use

Enzymes are proteins that catalyze chemical reactions. For example, in Bio 6B you'll use restriction enzymes to cut DNA molecules at specific location. Since the enzyme is a catalyst, it doesn’t matter exactly how many moles or micrograms of an enzyme you have; it matters how much DNA the enzyme can cut. Therefore, the concentration of the enzyme is measured in enzyme activity units. Each type of enzyme has its own defined unit of activity. For example, the restriction enzyme Eco RI has the following unit definition:

1 unit of enzyme is enough to completely cut 1 μg of lambda DNA in 1 hour at 37°.

When we buy restriction enzymes, they are labeled with a concentration like 50,000 Units/ml. If you want to cut 0.5 μg of lambda DNA with this enzyme solution, your calculation would look like this:

This gives you an answer in terms of enzyme units, but you need to know how many microliters to pipet. If you factor in the enzyme concentration (50,000 Units/ml in this example) and a conversion factor from milliliters to microliters, you get this:

This is a tiny amount, too small to pipet. Luckily, it won't do any harm to add some extra restriction enzyme. In the 6B lab, you should normally multiply the calculated amount of restriction enzyme by 10 to make sure you have enough. This is important because enzymes aren't very stable; if it was 50,000 Units/ml when it left the factory, it might be considerably less by the time you use it.

## Molecular weights of monomers and polymers

The molecular mass of macromolecules is often expressed in terms of daltons (Da). A dalton is more or less the same as the molecular weight or atomic mass units you learned in Chemistry. It is defined as follows: 1 Dalton = 1 gram/mole The molecular weight of a large polymer such as a nucleic acid or a polypeptide depends on the length of the polymer. The monomers (nucleotides or amino acids) vary slightly in molecular weight, but for a long polymer you can just use the average mass of a monomer, according to the list below:

• Double-stranded DNA (dsDNA) 618 daltons/nucleotide pair
• Single-stranded DNA (ssDNA) 309 daltons/nucleotide
• Single-stranded RNA (ssRNA) 320 daltons/nucleotide
• Polypeptide 110 daltons/amino acid (a rough estimate, since different amino acids vary widely in mass; two popypeptides with the same number of amino acids could have different molecular masses).

You can use this to estimate the molecular mass of a polymer based on the mass of its monomers. For example, the molecular weight of a polypeptide 100 amino acids long would be: (100 amino acids)(110 daltons/amino acid) = 11000 Da, or 11 kDa (kilodaltons). You could also use molecular mass (grams/mole) to calculate the mass of a single molecule: 1 Dalton = 1 gram/mole = 1.66 x 10-24 grams/molecule Once you have the molecular weight in Daltons, you can get the mass per molecule by dividing the molecular mass in Daltons by Avogadro’s number (6.02 x 1023 molecules/mole) or by multiplying by the reciprocal of Avogadro’s number, which is 1.66 x 10-24 mole/molecule.

### Proteins are in Da, DNA is in bp or kb

Daltons are the standard way of expressing the sizes of proteins. Most proteins are in the thousands of Daltons, so we usually give their sizes in kDa. For nucleic acids, daltons are less commonly used. Instead, most DNA and RNA molecules are simpy described in terms of base pairs (for double-stranded) or nucleotides (for single-stranded). If a double-stranded DNA fragment is 1200 nucleotides long, we could call it 1200 bp or 1.2 kb (kilobase pairs). One reason for for using kb instead of kDa is that every base pair of DNA has more or less the same mass, whether it's a G-C pair or an A-T pair. Another reason is we often cut or join DNA fragments; this is rarely done with proteins.

## How much DNA to load on a gel

When you perform DNA electrophoresis, you want to load an appropriate amount of DNA so you can clearly see the bands on the gel. If there's too much DNA, the bands will be smeared; if there's too little DNA, you won't see the bands. When you do a restriction digest, the smaller bands (shorter DNA fragments) are going to contain a lower mass of DNA, so the ability to see your smallest band will be a limiting factor for your experiment. In terms of seeing your results, the important thing is the mass of DNA present in each band. Usually the problem is having too little DNA, rather than too much.

For example, suppose you have a sample of plasmid DNA that you want to cut with a restriction enzyme and then analyze on a gel. Suppose that when you cut the DNA, you expect to get two fragments: 800 bp and 4000 bp. You might go through the following steps to figure out what to do.

#### What is the DNA concentration?

You have 200 µl of your DNA sample. You put 10 µl of DNA sample and 190 µl of working solution into an assay tube, and measure a concentration of 1.0 µg/ml in the assay tube. What is the DNA concentration in your sample tube? You can figure this out using C1V1=C2V2. In this case,

(V1)(10 μl)=(1 μg/ml)(200 μl)

Solve for V1; it's 20 µg/ml, which is the concentration of your DNA sample.

#### How much DNA do you need to use?

The limiting factor in this example will be having enough DNA in the small 800-bp band. The amount required depends on the staining method you use. For DNA gels in the Bio 6B lab, let's say that 50 ng DNA in a single band is ideal. If you're going to have 50 ng in the 800-bp band, you'll need to load more than 50 ng DNA in one lane of the gel, because some of the DNA will be in the 4000-bp band. In fact, since the 4000-bp DNA is five times as long as the 800-bp DNA, the larger band will contain five times as much DNA as the smaller one. So if there's 50 ng in the 800-bp band, there will be 250 ng in the 4000-bp band. You'll need to load 300 ng in one lane of the gel (50 ng + 250 ng). Using the DNA concentration above,

(300 ng)(1 µl/20 ng)=15 µl DNA

That's the amount you need to start with to give you 50 ng in the small band after you cut it.

#### How do you set up the restriction digest?

You know that you need 15 µl of DNA. For a restriction digest, you'll also need to add some restriction enzyme, buffer, and perhaps some water.

Since you've already figured out how much DNA to use, you can calculate the amount of enzyme as shown above. Calculate the theoretical minimum amount of enzyme, then multiply by 10 to make sure the DNA gets completely cut.

To calculate the amount of enzyme buffer, you need to know the final volume. This is actually easy, because you can choose the final volume. As long as it's large enough to contain your DNA, enzyme and buffer, but not too large to fit in a well of the gel (no more than 25 µl), any volume will do. Knowing this, you can now calculate the amounts for a complete restriction digest for the example given here.

You might see this presented as a quiz or lab final question. You won't necessarily be given the final volume. Even if the enzyme buffer is not mentioned, you should assume that there must be an appropriate buffer; it is usually supplied as 10x.

## Practice questions

I don't expect you to be able to do all these calculations at the beginning of the quarter, but I hope you can do them by the end.

1. 200 ng = ______ μg
2. 170 μg/ml = ______ mg/ml
3. Suppose you’re going to run a protein gel, and you need to dilute the running buffer. You have a stock solution that is 20x, and you want to make 500 ml of 1x buffer. How much of the 20x stock solution should you use (in ml)?
4. Suppose you have a DNA solution that is 800 ng/μl, and you want to dilute some of it to make 100 μl of a solution that is 80 ng/μl. How much of your original DNA solution should you use, and how much water should you add?
5. Suppose you have a DNA solution that is 800 ng/μl, and you want to use 0.01 mg of DNA in a reaction. How many microliters of your DNA solution should you use?
6. Suppose you have a sample of plasmid DNA that you want to analyze by performing a restriction digest and a gel. You have 200 μl of your DNA sample, and the DNA concentration is 40 μg/ml. The uncut DNA is 10 kb long, and you expect it to be cut into fragments of 6 kb, 3 kb, and 1 kb. You want to cut enough DNA to give you 50 ng in the 1-kb band, and then load it all on the gel. Show your calculations for everything you would put into the restriction digest tube. The stock concentration of the restriction enzyme is 5 Units/μl, and one unit of enzyme cuts one μg of DNA.

## More resources

### Online lab calculators

There are some good online calculators and apps for the molecular biology lab. I encourage you to explore these; you can use them in lab, but you can't use them on quizzes or exams. These calculators keep track of the units for you.

Tocris Dilution Calculator

Stock Solution Calculator from EasyCalculation.com

NEB (New England Biolabs) Interactive Tools. NEB is a biotech company; we buy some of our enzymes from them. They have some useful online calculators as well as an app (search the app store for it).

### General scientific calculators

In case you don't have a calculator handy, here are a couple of online scientific calculators. These just do numbers, not the units.

### Videos

Calculating Dilutions in the Lab from Synthetic Biology One. Explains the rationale for lab calculations. A related video from the same organization: Common Units of Concentration in the Lab.