# A Guide for Solving Your Lab Math Problems

Math is an important part of lab life, from making solutions to calculating protein concentrations, and miscalculations can cause mayhem for your experiments. Therefore it is important that your math is right, or you could spend weeks trying to figure out what’s going wrong in your experiments.

I was hopeless at remembering how to do even simple calculations, so I kept a cheat sheet in the back of my lab book that I referred to on a regular basis.

I want to make your life easier too, and that’s why I’ve put together some of the key (in my opinion) calculations important for a molecular biologist.

## Making up solutions

The routine chore that everyone avoids until absolutely necessary. There are three key equations that you will need in order to make up simple solutions.

1.  Calculating moles. If you need to make up a solution where you know the desired concentration (molarity) and volume then you first need to calculate the number of moles in that solution.

The calculation for this is simple:

n = M x V

That is:  moles = Molarity (concentration in molar)  x volume ( in litres)

2.  Once you’ve got the moles you can then work out the mass required using the following equation:

m = n x Mw

Where:  mass (in grams) = moles x molecular weight ( in g mol-1).

3.  The two above equations will enable you to make solutions from scratch but what if you want to make a solution where you already have a stock solution of a higher concentration?

Diluting stock solutions is really simple and can be achieve using the following calculation:

V1 x C1 = V2 x C2

V1 =Volume of stock solution required

C1 =Concentration of stock solution

V2 =Volume of final solution

C2 =Concentration of final solution

So in the case where we need to find the volume required from our stock solution we rearrange the equation so that:

V1 = (V2 x C2) / C1

The units aren’t important except that the volumes and concentrations must be in the same units.

## Calculating concentrations of DNA or RNA

The simplest way to calculate DNA or RNA concentration is using a spectrophotometer. For a 1 cm path length (this is the width of the cuvette – most cuvettes have a standard width of 1 cm)  dsDNA at a concentration of 50 µg/mL and RNA at a concentration of 40 µg/mL has an optical density of 1 at 260nm. This means by measuring the optical density of a solution of DNA or RNA at 260nm we can determine the concentration in our sample using the following calculations:

ds DNA concentration (in µg/mL) = 50 x OD260 x dilution factor

RNA concentration (in µg/mL) = 40 x OD260 x dilution factor

The dilution factor is the dilution of the solution measured compared to the ‘stock’ solution of your DNA. It’s a good idea to dilute your DNA for measuring the OD for several reasons; firstly so you don’t use up all of your stock; secondly DNA can be quite viscous at high concentrations which can introduce pipetting errors and finally you want to aim for a OD between 0.1 and 1 in order to get the most accurate quantitation (this is the linear range for most spectrophotometers). As 260nm is in the UV spectrum you need to use a specialised UV cuvette.

## Calculating purity of DNA or RNA

As much as we all like to feel like perfectionists in the lab, solutions of DNA and RNA contain contaminants that can affect the optical density at 260nm. Therefore it’s a good idea to test the purity of your DNA and RNA by measuring for contaminants. The main contaminant of extracted DNA and RNA is proteins, which also absorb at 280nm. To check for the presence of contaminating proteins in your sample simply measure the OD at 280 nm and calculate the ratio of OD260/OD280. Pure DNA should have a ratio of ~2 while pure RNA should have a ratio of ~1.8.

There are other contaminates which can also be measured at different optical densities to determine how pure your sample is, but proteins tend to be the main culprits.

## Converting units

Sometimes it’s easy to forget how many picograms are in a gram so it is handy to have a simple reference to check just to make sure. The easiest way to remember, is that the difference between most units is 103;  or just check out the handy table below:

PrefixFactorSymbol
Kilo103k
Deci10-1d
Centi10-2c
Milli10-3m
Micro10-6$\mu$
Nano10-9n
Pico10-12p

So in order to convert picograms to grams you need to multiply by 10-12.

## Ligation calculations

For an optimal ligation reaction you want a 1:1 ratio of insert to vector, although alternative ratios can be tested (such as 1:3, and 3:1). To calculate the amount of insert required for a 1:1 ratio use the following equation:

(kb insert / kb vector) x ng of vector = ng insert required

## Calculating cell concentration using a haemocytometer

Seeding cells at the correct density is important for the consistency of experiments as well as to maintain a healthy stock of cells. Calculating cell concentration is simple with the use of a haemocytometer. Simply apply the cell solution to the haemocytometer and count the number of cell in a 1mm x 1mm square (made up of 5×5 small squares).

Then use the following calculation:

number of cells x dilution factor x 104 = Cells per mL

I would usually use a dilution of 1:1 with trypan blue (a vital stain that will stain any dead cells blue), but you can alter this depending on how concentrated your sample is. You want to count roughly between 40 and 70 cells in order to get an accurate reading.

Haemocytometer grid.

For a quick reference guide download and print out my simple cheat sheet here to ensure all the necessary calculations are easy at hand.

Image Credit: Challot

1. Ludwig on December 27, 2018 at 1:37 am

You wrote:
“To check for the presence of contaminating proteins in your sample simply measure the OD at 280 nm and calculate the ratio of OD260/OD280. Pure DNA should have a ratio of ~2 while pure RNA should have a ratio of ~1.8.”

But DNA should be around 1.8 and RNA around 2

2. Kingz on June 7, 2016 at 1:13 pm

HI, if I have v1=1ug x 10ul/500ug/ml wht will my final si unit be for the answer? sorry I’m just slow in remembering simple maths protocols
Thanks a lot for the above information much helpful.

3. Pranay Dogra on December 17, 2013 at 10:56 pm

Thank you for summarizing the important calculations. Could you also add the calculation to determine the number of molecules in a sample of DNA (plasmid and PCR)?

I also noticed a typo in the article above under “Calculating cell concentration using a haemocytometer”. I think that the formula should read 10^4, where as you have it as 10^-4.

Thanks again.

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