poisson distribution

This weekâ€™s question comes from a graduate student studying virology, who writes:

My professor recently said that really, the MOI doesn’t matter in a culture, it is the concentration of viral particles in the media that matters. Ie: if you have 10 million cells or one cell, but you are infecting the plate with 5mL of 100 million viral particles/mL, then the amount of virus interacting with each cell is not different in either scenario (pretending that it isn’t nearly impossible for that single cell to survive in culture alone). I argued with him, saying that the cytotoxicity to the single cell would certainly be increased. He then said that a student hadn’t argued with him about that in his 15 years of teaching and I promptly decided to get some evidence before I continued the discussion.

I’m not actually sure which side is correct. I know that concentration is certainly a large determinant for infectious events/cell. But, it is hard for me to understand why MOI wouldn’t be more important? The more I think about it the more I think that I may be wrong. But if you have two plates with equal numbers of cells, and you add 5 mL of media to one and 50mL of media to the other – assuming that the media is 100 million infectious particles/mL – would the higher MOI plate not result in more infectious events per cell?

My reply: What first jumps outÂ at me is the fact that the professor is using the no one ever argued with me about thatÂ excuse to sayÂ that he/she is right. That is the exact role of a student, to ask questions, and it should never be discouraged. Students can ask the best questions because they are frequently unencumbered by the bias of a field.

Please tell your professor that both multiplicity of infection and concentration of viral particles matter, for different reasons. The multiplicity of infection (MOI) is the number of virus particles added per cell. If you add one million virus particles to one million cells in a culture plate, the MOI = 1. If you add ten million virus particles to one million cells, the MOI is 10.

However, if one million virus particles are added to one million cells, each cell will not be infected with one virus particle. How many cells are uninfected, or receive 1, 2, or more virus particles is determined by the Poisson distribution. At an MOI of 1, 37% of the cells are uninfected, 37% receive 1 particle, 18% receive 2 particles, and so on.

In theory, the number of particles that infect each cell is controlled by the MOI, not the virus concentration. However, when the concentration of virus particles is very low, attachment to cells will take a very long time. This is becauseÂ virus attachment is governed by the concentrations of free virions and host cells. The rate of attachment can be described by the equation

dA/dt = k[V][H]

where [V] and [H] are the concentrations of virions and host cells, respectively, and k is a rate constant.

For a 6 cm culture dish with an area of 113 square cm, we typically infect with virus in a volume no greater than 0.1 – 0.2 ml. In this way virus attachment to cells will be essentially complete within 1 hr at 37 degrees C. If the same amount of virus were added in 10 ml of medium, the attachment would take much longer; however because the MOI is the same in both cultures, at the end of the adsorption period the number of infected and uninfected cells in both cultures would be the same.

To answer the readerâ€™s last question:

But if you have two plates with equal numbers of cells, and you add 5mL of media to one and 50mL of media to the other – assuming that the media is 100 mill infectious particles/mL – would the higher MOI plate not result in more infectious events per cell?

The answer is yes – assuming you wait long enough for the viruses in the more dilute culture to attach to cells.

Multiplicity of infection (MOI) is a frequently used term in virology which refers to the number of virions that are added per cell during infection. If one million virions are added to one million cells, the MOI is one. If ten million virions are added, the MOI is ten. Add 100,000 virions, and the MOI is 0.1. The concept is straightforward.

But here is the tricky part. If you infect cells at a MOI of one, does that mean that each cell in the cutlure receives one virion?

The answer is no.

Here is another way to look at this problem: imagine a room containing 100 buckets. If you threw 100 tennis balls into that room – all at the same time – would each bucket get one ball? Most likely not.

How many tennis balls end up in each bucket, or the number of virions that each cell receives at different MOI, is described by the Poisson distribution:

P(k) = e^-mm^k/k!

In this equation, P(k) is the fraction of cells infected by k virus particles, and m is the MOI. The equation can be simplified to calculate the fraction of uninfected cells (k=0), cells with a single infection (k=1), and cells with multiple infection (k>1):

P(0) = e^-m

^{P(1) = me^-m}

^{P(>1) = 1-e^-m(m+1)*}

^{*this value is obtained by subtracting from unity (the sum of all probabilities for any value of k) the probabilities P(0) and P(1)}

Here are some examples of how these equations can be used. If we have a million cells in a culture dish and infect them at a MOI of 10, how many cells receive 0, 1, and more than one virion? The fraction of uninfected cells – those which receive 0 particles – is

P(0) = e^-10

= 4.5 x 10^-5

In a culture of one million cells this is 45 uninfected cells. That’s why an MOI of 10 is used in many virology experiments – it assures that essentially every cell is infected.

At the same MOI of 10, the number of cells that receive 1 particle is calculated by

P(1) = 10e^-10

= 10 x 4.5 x 10^-5

= 4.5 x 10^-4

In a culture of one million cells, 450 cells receive 1 particle.

How many cells receive more than one particle is calculated by

P(>1) = 1-e^-10(10+1)

=0.995

In a culture of one million cells, 999,500 cells receive more than one particle.

Using the same formulas, we can determine the fraction of cells receiving 0, 1, and more than one virus particle if we infect one million cells at a MOI of 1:

P(0) = e^-1 = 0.37 = 37% of cells are uninfected

P(1) = 1 x e^-1 = 37% of cells receive one virion

P(>1) = 1 – e^-1(1+1) = 26% of cells are multiply infected

An assumption inherent in these calculations is that all cells in a culture are identical in their ability to be infected. In a clonal cell culture (such as HeLa cells) the deviations in size and surface properties are small enough to be negligible. However, in a multicellular animal there are substantial differences in cell types that affect susceptibility to infection. Under these conditions, it is experimentally difficult to determine how many virions infect different cells.

High MOI is used when the experiment requires that every cell in the culture is infected. By contrast, low MOI is used when multiple cycles of infection are required. However, it is not possible to calculate the MOI unless the virus titer can be determined – for example byÂ plaque assay or any other method of quantifying infectivity.