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Serial Dilution
Dec 2010
page 1
Population Dynamics:
Serial Dilution
URL:
http://mathbench.umd.edu/modules/cell-processes_serial-dil/page01.htm
Note: All printer-friendly versions of the modules use an amazing new interactive technique called
“cover up the answers”. You know what to do…
Recap the story
Remember Frank? Frank who had roommate from Sweden? Frank, who accidentally managed to
host a few super-virulent meningicocci -- the kind that are common in Sweden, but rare here?
Frank, who is now sitting in the doctor's office, looking a bit like one of those angsty teenage
vampires, with a jacket over his head to protect his eyes, a rash all over his body, and a neck as
stiff as a board? Yeah, that Frank.
Well anyway, when the young medical intern (you!) cultured some meningicocci, you found that
they double just about every half hour. If Frank's body is as nice a place for meningicocci as the
petri dish is, you decided Frank should be dead by now. Poor Frank...
Ah, but luckily Frank's body is not quite as cushy as a petri dish, due to his killer immune
system, and also the inconvenient fact (if you're a meningicoccus, that is) that Frank's blood
keeps its iron safely tucked away inside red blood cells. Curses, foiled again!
Nevertheless, even though Frank's not dead, he's not doing so well either. And what we need to
do is to find out just how badly off he really is. We need to measure that infection. But how?
Count 'em all?
Let's get one thing out of the way, right away. Given that Frank has been sick for a half day or
so, there could be millions, even billions, of meningicocci in his blood. Do we want to count
every single one?
Yes, I love to count: Hmm, try again.
No, because that would mean filtering all of his blood: That's true, and there's another reason as well.
No, because it takes a really long time to count a billion: Yes, this is a really good reason not to
count every meningicoccus.
So, having established that we can’t count’em all, what else can we do?
We could sample a little of his blood: Yes, we could. We should. We will.
We could give up: Hmm, I don't think that's the way into med school...
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Serial Dilution
Dec 2010
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Direct count
So, Frank's got about 5 liters of blood, and he probably won't miss a few mLs all that much. Let's
say we took that 10 mL, centrifuged out the gigantic stuff like red blood cells, put the rest on a
slide under a microscope, and counted every thing that looked like a meningicoccus. That's
certainly one way of quantifying the population. We'll call it ... Direct Count. Pretty easy,
conceptually speaking, although it takes a bit of time to do the centrifuging and to make the
slide, and even more time to actually count. But, that's what underpaid lab interns are for, right?
OK, so you've counted using a Petroff Hausser chamber under the light microscope and found
that you have 326 meningicocci in 10mL of blood. What does that mean?
A less experienced lab intern might run off to tell the doctor that Frank has a total population of
326 meningicocci living in him. Not you, however, because you remember that you only
sampled 10 mLs of Frank's blood.
What you need to know is what fraction of Frank's blood you sampled:
10mL/5000mL = 1/500th
So, in Frank as a whole, you would expect to find about 500 times as much as you did in your
sample.
500 * 326 = 163,000 meningicocci
This scaling up method (sometimes called multiplying by the inverse) is absoluately critical
for successfully calculating the actual population from a sample.
The online version of this module contains an interactive
applet which allows you to practice scaling up. To find this
applet go to: http://mathbench.umd.edu/modules/cellprocesses_serial-dil/page03.htm
Light scatter (using spec)
Direct count is a straightforward way to figure out how many meningicocci Frank is harboring,
but it is a bit labor intensive. Wouldn't it be nice to get a machine to do the counting for us?
One way we can do this is to use a photospectometer. A spec doesn't exactly count the cocci,
but it does measure how much they interfere with a light beam, and based on that, we can
use a calibration to decide how many cocci we've got. Then we can do the scaling up trick
to get the total population size.
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Serial Dilution
Dec 2010
page 3
This is a great method if you already have a calibration curve -- perhaps you work in a lab that
has been studying this critter for years... Otherwise, you may need to spend a couple of days
doing the calibration (growing the cultures, taking the spec readings, comparing them to counts
under the microscope... that sort of thing).
Here's an example of a calibration curve:
For example, if you prepared a sample and got a spec reading of 0.5, you would find 0.5 on the
x-axis and read the bacterial count off the y-axis -- about 26 million
look at the graph!
How about a spec reading of 0.9? (Be careful with the log scale for bacteria!!)
about 75 million
If your sample had about 12 million bacteria, what OD reading would you expect?
OD = 0.2
The dilemma of the dead cell
So far we have one hard way (Direct Count) and one easy way (Photospec) to quantify Frank's
meningicoccal load. But both ways share a big problem ... the dilemma of the dead cell.
If you were counting, say, cats ... this wouldn't be a problem. A dead cat and a live cat are
generally not easily confused. If in doubt, you could poke the cat, and you would know for sure.
However, bacterial cells are much less lively than cats. Yes, there is plenty of stuff going on
inside, but in general you're not going to see that from the outside. Under a microscope, cells
that are alive and cells that are dead look a lot alike. And to a spec, live and dead cells look
exactly alike.
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Serial Dilution
Dec 2010
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Based on the dilemma of the dead cell, which would you predict?
Direct count and photospec will...?
Overestimate the population: yes, these methods overestimate, because the dead cells get counted as
if they are alive
Underestimate the population: the population won't be underestimated, because too many cells get
counted, not too few.
Get the population just right: the population will not be accurately estimated, both methods count
too many cells.
If we care about getting the number of live cells right, we need a better way…
Viable plate count
So we can't poke a cell to see if it's alive -- how else can we make the call? What do live cells do
that dead cells don't? Well, one big thing is, live cells grow and reproduce. And they do it fairly
quickly (remember, every half hour or so in the case of meningicocci). So if we were to take a
batch of cells and wait a day or so, we would soon know if they were alive or not.
Let's refine this idea a little, by following... A day in the life of a microbe
Out in the real world (like in your nose, throat, or bloodstream), meningicocci are limited by
temperature and nutrients. In the lab, we're going to put the meningicocci on a media, which will
do two things:
1. take away limiting factors, and
2. fix each individual cell in one place, so it doesn't get moved around.
So, let’s watch a single meningicoccus cell, conveniently named Minnie, sitting in the middle
of a batch of media. The media has every creature comfort that Minnie could want, and she pulls
in sugar as fast as she can. Soon she’s grown out of her membrane and she splits in half,
creating little Minnies 2 and 3. These two also pull in sugar as fast as they can, and a half hour
later, they too start to feel ready to split, so they do. Minnies 4, 5, 6 and 7 keep going.
time: 0 min
time: 50 min
time: 1 hr 40 min
time: 2 hrs 40 min
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Serial Dilution
Dec 2010
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Luckily, none of the Minnie's move around much. Although some bacteria can move over the
surface of agar, meningicocci are good at doing this. So, they’re all stuck sitting in a little pile,
but they don’t mind, as long as there’s plenty of sugar to go around. And pretty soon we have a
pile consisting of Minnies 8 through 15, all descendents of Minnie the First. And so on.
Eventually (after about 24 hours) this pile will be big enough for one of us hulking human
beings to see without even using a microscope. This tiny but visible pile is called a colony,
and the original Minnie was the original colony-forming unit, or CFU.
Of course Minnie was probably not the only CFU around. Over on the other side of the agar,
Ginnie was sitting around, minding her own business, enjoying the warmth and sugar, and
growing at the same rate. So while a few million descendants of Minnie formed a colony on one
side of the plate, a few million descendants of Ginnie form a colony of the other side. And so
on, one colony for each original colony forming unit.
Comparison of methods
We've talked about 3 methods so far: just plain counting under a scope, using a
photospectometer to do the dirty work, or growing bacteria in a dish to count the piles of
descendants they leave.
There is one other way that we could (theoretically) figure out how many cells there are: we
could gather them all in one place and weigh them. This would be the biomass method. Biomass
is an important way of measuring populations in the lab, but in Frank's case it would not help us
-- we would need to collect about 10 million bacterial cells before we could weigh them, and if
we could do that, we could probably just cure Frank then and there.
So, let's compare the 3 practical ways we have of measuring the infection:
Which one requires
the least time (for
example, the boss
wants the answer
before lunch)?
Which one requires
the least effort (think
lab intern pain)?
Which one can
distinguish live from
dead cells?
And one more question:
Direct Count
NOT TOO BAD -although counting
can be timeconsuming
Spec
FASTEST -- but only if
you already have the
calibration curve!!!
Viable Plate Count
SLOWEST -- we
have to wait an entire
day for the cells to
grow
UGH -- counting
under a scope gets
old fast!
WOO-HOO -- very easy to
do, even if you didn\'t
understand the math -unless of course you're the
one that has to make the
calibration curve
SORRY, you're outa luck
SOME WORK -- but
most people prefer it
to counting under a
scope.
NOPE, no can do.
YES!! That's the
genius of this method.
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Serial Dilution
Dec 2010
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One day an overzealous intern decided to compare all 3 methods. Here's what she found,
but she forgot to label which reading came from which method:
347 organisms/mL
520 organisms/mL
510 organisms/mL
Can you tell which reading is which? And what phase of growth (lag, growth, stable,
death) is the population in?
Which is which? The first reading is definitely the viable plate count, which doesn't include
dead cells. And no, you can't really tell the difference between the other two.
Lag, growth, stable, death? The population is probably in the death phase, since there are a lot
of dead cells around.
Why we need to dilute
For the remainder of this module, we're going to stick with the Viable Plate Count method -the only method that can tell a live cell from a dead one. But we still have another problem -- the
possibility of lots of lots of colonies to count. For example, 500 colonies on a petri dish would
look something like this:
What if you are trying to count a population in the thousands or millions? You could literally
have a carpet of colonies (also known as a 'confluent lawn') growing on your petri dish. This is
where dilution saves the day. Not just dilution, but serial dilution... meaning dilution over and
over again.
Why do we dilute? To have less to count
Why do we do it repeatedly? Because we don't know how much dilution we need. Every time
we dilute, we'll also make a new plate to incubate. So we might do 5 dilutions and grow up 5
plates. Then we'll end up throwing away 4 of them. Sound wasteful? Well, dilution and plating is
quick and easy compared to the pain of starting your experiment all over again.
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Serial Dilution
Dec 2010
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Interlude: Diluting Cold Press Coffee
So let's see what the process of "diluting" looks like. Here's an everyday example:
Cold Press coffee is a superstrong form coffee made without heating the water. The brew starts
out with about a zillion molecules of caffeine per cup. In order to actually be able to drink the
stuff, you have to dilute it. For example, you could put one cup of cold press coffee into a pot
and add 4 cups of water.
How many molecules of caffeine are in the pot? a zillion
How many cups of coffee are in the pot ? 5 (1 cup of cold press + 4 cups of water)
If you pour 1 cup from the pot, how many molecules of caffeine will it have?a fifth of a zillion
That would work for a true coffee aficionado, but many of us are less hardy (or more wimpy) so
we might want another dilution. Let's take 1/2 cup from the pot, and add 4.5 cup of water to
make a wimpy pot.
How many caffeine molecules were in the 1/2 cup? 1/2 cup is a tenth of the entire pot, so there
should have been one-tenth of one-fifth of a zillion caffeine molecules in the half cup.
How many caffeine molecules are in our final (wimpy) pot of coffee? all the caffeine from the
half-cup goes into the wimpy pot, and no more get added from the water, so again, onetenth of one-fifth of a zillion.
The Sunbucks across the street makes a wimpy coffee with exactly 13,000 caffeine
molecules in a cup. Assuming they used the same dilution scheme, how many caffeine
molecules were 1 cup of the original brew?
Here's the dilution scheme:


1 cup brew + 4 cups water  5 cups strong coffee, then
0.5 cup strong coffee + 4.5 cup water  5 cups wimpy coffee
Another way of saying this is,



The first time we diluted, we kept 1/5 of the caffeine from the brew
The second time we diluted we kept 1/10th of the caffeine from the strong coffee
1/5th of 1/10th = 1/5*1/10 =1/50.
So, working backwards, the original brew must have contained 50 times as much
caffeine, or

50*13,000 = 650,000 molecules caffeine / cup
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Serial Dilution
Dec 2010
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The dilution factor
When you're thinking about dilution, it helps to simplify your actions into dilution factors. For
example, if you
1. 1 mL coffee + 4 mL water = 1 in 5, or 0.2 dilution factor
2. 1 mL coffee + 9mL water = 1 in 10, or 0.1 dilution factor
3. 1 mL + 99mL water = 1 in 100, or 0.01 dilution factor
As you've probably guessed, this works exactly the same whether you're talking about caffeine or
meningicocci. Here's what a dilution factor of 0.01 looks like on the lab bench:
Notice that it really doesn't matter how much of the original stock you started with, as long as
you had at least a mL to put into the new container. What matters is how much you transfer and
how much water you add. The dilution factor is then defined as
amount transferred / total amount =
amount transferred / (amount transferred + amount water added) =
1 / (1+99) = 1/100 = 0.01.
Overall dilution
What if you do dilute and then dilute again? For example, in our cold press example, the original
dilution factor was 1/5, and the second dilution factor was 1/10. So,
strong coffee had 1/5th of the caffeine in the brew, and
wimpy coffee had 1/5th of 1/10th = 1/50th of the caffeine in the brew.
As you can see, it makes sense to multiply the dilution factors to find the overall dilution factor.
This works regardless of how many dilutions you do.
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Serial Dilution
Dec 2010
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overall dilution factor = dilution factor 1 * dilution factor 2 * dilution factor 3 * ...
Over at Sunbucks, the manager would like to make a kids version of the drink that will
only have 1/100th of the caffeine of the original brew. How can he do this?
He could take the wimpy coffee (dilution factor 1/50) and apply a dilution of 1/2, since
1/50 * 1/2 = 1/100.
And how can he achieve a dilution factor of 1/2?
He could transfer 1 cup of wimpy coffee and add 1 cup of water.
Or if he had a big group coming, he could transfer 6 cups of wimpy coffee and add 6
cups of water.
Or there are lots of other ways of making a 1/2 dilution.
Could he make the kids drink in some other way (not involving a 1/2 dilution)?
Sure, he could start with the strong coffee (1/5 dilution) and make a second dilution:
1/5 * 1/20 = 100. (One cup coffee plus 19 cups water)
Or he could do it in two extra steps:
1/5 * 1/5 * 1/4 = 1/100.
As you can see, there are lots of ways of getting to the same overall dilution.
Back to the lab: (PIC to illustrate)
We start with a concentrated stock of meningicocci virus and perform the following dilutions:
Add 1 mLto 9 mL water: Dilution factor 1/10 = 0.1
Then, add 1 mL to 99 mL water: Dilution factor 1/100 = 0.01
Then, add 1 mL to 99 mL water: Dilution factor 1/5 = 0.2
What was the overall dilution factor?: 1/10*1/100*1/5 = 1/5000 = 0.0002
As a way to double-check, you can think the dilution process through logically. Often it helps to
think through the whole process using some concrete number. For example, if you did two
1/100 dilutions, you could think about it like this: "I started with 300,000 cells in a mL, and put
them into 99 mLs of saline, and took out a mL, so there must have been 3,000 cells in that mL.
Then I put those 3,000 cells into 99 mLs of saline, and took out one mL again, so there must
have been 30 cells in that mL. Overall my dilution must have been to 30 from 300,000, which is
the same as to 1 from 10,000 (1:10,000)." Of course, the idea that you started with 300,000 cells
is pure fiction, but it can help you make sure that you've done the dilutions correctly.
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Serial Dilution
Dec 2010
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How to scale back up
"Scaling up" means starting from a sample and figuring out how many were in the original
brew, or stock, or whatever was originally there.
For example, we can start with a cup of wimpy coffee and figure out how much caffeine was in
the original brew. All we need to know is what the overall dilution factor was. In the case of
wimpy coffee, it was 1/50, or 0.02.
So, when we count the caffeine molecules in a cup of wimpy coffee, we know we got 1/50th of
what was in a cup of the original brew -- or in other words, there was 50 times as much in a cup
of the original. This method is called multiplying by the inverse (of the dilution factor).


If the dilution factor is in the form of a fraction, you can "flip" the fraction (i.e., 1/50 becomes
multiply by 50/1).
If the dilution factor is in decimal form, multiply by 1 over the decimal (i.e., 0.02 becomes
multiply by 1/0.02).
Finally, notice that I'm telling you the total number of caffeine molecules ONE CUP of the
original brew. If I don't know the actual amount of original brew, I also won't know how much
caffeine there was total.
Back to the lab
Here are some problems, ranging from easy to a bit hard...
I did a series of dilutions with an overall dilution factor of 1/20,000, and then plated a
grew a 1mL sample. After 1 day, I counted 27 CFUs on the petri dish. How many CFUs
would there be per 1mL of the original stock?


Inverse of 1/20,000 is 20,000
27*20,000 = ...
Answer: 540,000 per mL
I did a series of dilutions, with dilution factors of 0.1, 0.1, and 0.01. At the end I plated and
grew a 1mL sample, and counted 48 CFUs. How many CFUs would there be per 1mL of
the original stock?




What's the overall dilution factor?
Overall dilution factor = 0.1*0.1*0.01 = 0.0001
Multiply by the inverse...
Overall dilution factor = 0.1*0.1*0.01 = 0.0001
Answer: 48 * 1/0.0001 = 480,000 CFUs per mL
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Serial Dilution
Dec 2010
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I did a series of dilutions as follows:
*1 mL added to 9 mLs water
*1 mL added to 99 mLs water
*1 mL added to 49 mLs water
If the final 1mL sample had 152 colonies, what was the original concentration?


The dilution factors were 1/10, 1/100, and 1/50
The overall dilution was 1/50,000
Answer:152 * 50,000 = 7,600,000 cells/mL
Why we plate more than one dilution
Remember the picture of the hell-to-count petri dish with 500 colonies? No one wants that to
happen to them, especially late on a Friday afternoon. But on the other hand, how can we avoid
it?
Very easily ... while we're doing our dilutions, we just keep plating each intermediate step. Then
the next day, we decide which plate looks like the most reasonable for counting. Since we
cleverly labelled each plate with its overall dilution factor, we also know how to scale back up to
get the original concentration.
So, now we finally have all the pieces of serial dilution assembled:




Dilute
Calculate dilution factor
Plate
Repeat
To make things easier, the standard operating procedure is to go by factors of 10, and to do
about 5 or 6 plates altogether.
The online version of this module contains an
interactive applet which allows you to look at
different serial dilutions. To find this applet go to:
http://mathbench.umd.edu/modules/cellprocesses_serial-dil/page13.htm
So, for example, if I believe that I have about 42 million CFUs per mL, I might want to start with
a first dilution of about 1/1000, and plate that. Then I make 4 more 1/10 dilutions, plating each
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Serial Dilution
Dec 2010
page 12
one in turn. In the end, I have 5 plates, labeled as shown below. And, if you roll over the image,
you will see the plate counts I expect if my original guess (42 million) was correct
Picking the plate count
For this, we have any easy rule of thumb: the plate you count should have between 30 and 300
colonies.
This rule of thumb has the undeniable advantage of being easy to use. But why does it work?
Let's imagine that we have a sample with exactly 183,000 cells per mL. And let's say that our
technique is flawless. If we count a 1:1000 diluted plate, we should find 183 CFUs (remember,
we have magically flawless technique). But the exact same thing would happen if our sample
started with 183,001 cells. Or 183,100 cells. Or 183,400 cells. Or 182,800 cells. In fact, even
with our absolutely flawless technique, the best we can say is that there are between 182,500 and
183,500 cells. So our absolutely unavoidable error is plus or minus 500/182,000 -- about 0.27%.
That's pretty good.
Let's look at some other ways we could count the same 183,000 cell sample:
Dilution Factor
Plate count
Resolution
Ease of counting
1830
182,950 to 183,050
+/- 50 = 0.027%
great!!
182,500 to 183,500
+/- 500 = 0.27%
still pretty good
175,000 to 185,000
+/- 5000 = 2.7%
not too good
150,000 to 250,000
+/- 50000 = 27%
ACK!
NOPE, can't do it, this is
too much to count
1:100
183
1:1000
18
1:10,000
2
1:100,000
Boring
but count-able
Easy to count
Chimpanzees can count
this plate!
The more we dilute, the easier the counting, but the higher the error. So you can see that the 30
to 300 guideline is really a compromise between countability and error.
Putting it all together
The online version of this module contains an
interactive applet which allows you to quantify a
bacterial population. To find this applet go to:
http://mathbench.umd.edu/modules/cellprocesses_serial-dil/page15.htm
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Serial Dilution
Dec 2010
page 13
More dilution problems
The online version of this module contains an
interactive applet which allows you to practice
different dilution schemes. To find this applet go
to: http://mathbench.umd.edu/modules/cellprocesses_serial-dil/page16.htm
And one problem you'll really need to think about...
Starting with a dilution made by a TA, you add 1 mL to 49 mLs of water to get a 1:500,000
dilution. What was the TA's dilution factor?



Your dilution factor is 1/50
The overal dilution factor was 1/500,000
1/? * 1/50 = 1/500,000
Answer: 1/10,000
Quickies
Below, you can practice a couple more times. I also included a few unusual situations ... things
do go wrong, of course, and you should be able to recognize that. If you think the dilution
series is not valid, click on "re-do the dilution". Oh, and "TNTC" means "too numerous to
count".
Dilution
factor
1:100,000
TNTC
1:1,000,000
674
1:10,000,000
68
1:100,000,000
7
Count
Answer
Nope! TNTC
means "too
numerous to
count" !
Nope! 674 is
definitely more
than 250!
Yes -- And the
answer is ...
about
680,000,000
cells/mL
Nope! 7 is too
few to be valid
No, each count
is about 10th of
the count before
it, so the
dilution looks
good.
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Dilution
factor
1:100,000
Serial Dilution
1:1,000,000
Dec 2010
1:10,000,000
1:100,000,000
1
375
105
11
Nope! 375
is too many
to count
Something is
wrong! The
second count
dropped only by a
factor of 3, rather
than a factor of 10
Nope! 11 is too
few to be valid
page 14
Count
Answer
Dilution
factor
1:100
TNTC
1:1000
1:10,000
Nope! 1 is too
few to be valid
Yes, you should
redo the dilution
since the second
plate seems to
have too many
CFUs.
1:100,000
249
24
0
Nope! TNTC
means "too
numerous to
count" !
Yes -- and the
answer is
249,000
cells/mL
No -- 24 is
too few to be
valid
Nope! 0 is not
valid for
scaling up
1:10
1:100
Count
Answer
Dilution
factor
954
96
1:1000
10
No, each count is
about 10th of the
count before it, so the
dilution looks good.
1:10,000
0
Count
Answer
Nope! 954 is
definitley more
than 250!
Yes -- The
answer is
9,600
cells/mL
Nope! This is
too few to be
valid
Nope! 0 is
not valid
No, each count is about
10th of the count
before it, so the
dilution looks good.
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Dilution
factor
Serial Dilution
1:10,000
1:100,000
1:1,000,000
TNTC
567
58
Nope!
TNTC
means
"too
numerou
s to
count"!
Nope! 567
is
definitely
more than
250!
Yes -- although
it looks like
there was a
problem with
the next
dilution. The
answer is ...
about
58,000,000
cells/mL
Dec 2010
1:10,000,000
1:100,000,00
0
12
1
page 15
Count
Answer
Nope! 12 is
too few to be
valid
Nope! 1 is not
valid
No, although
it looks like
something
happened to
the 4th plate,
it doesn't
affect the
plate you
will use.
Is he dead yet?
Remember Frank? He's still sitting on the examining table with a coat over his head. In fact
about 50% of people with meningitis will die when the bacterial levels in their blood stream
reach 500 / mL.
The worried nurse hands you a vial of 10 mLs of Frank's blood. How can you decide whether the
infection is at a lethal level?
What counting method will you use?
Unfortunately, a direct count or a photospec would require separating the meningicocci from the
red blood cells. If we could do that, we could probably just cure poor Frank. So we're stuck with
a viable plate count.
What dilution scheme will you use?
We need to get plate counts between 30 and 300, and we think that the concentration in his blood
might be as high as 500 per mL, but it could be much lower. So, we should go ahead do the first
plate without even diluting! After that we could do standard 1/10th dilutions.
How many dilutions do you need?
You definitely need at least 1 dilution, in case Frank is somewhere in the 300 or 400 per mL
range. Without a dilution, that wouldn't be countable.
But really, a single 1/10th dilution would suffice. If Frank has >300 colonies in a 1/10th dilution,
he has >3000 actual cells per mL, and that would be fatal. (Of course, he'd also be dead, but let's
not be morbid).
www.mathbench.umd.edu
Serial Dilution
Dec 2010
page 16
However, you could keep going with a standard 5 dilutions and not waste much time or energy,
plus it would allow you to make an accurate estimate regardless of his actual level of
meningicocci in the blood.