A Fluctuation Test in Bacteria - University of Missouri ...faculty.missouri.edu/~glaserr/prism/BacterialFluctuationTest.pdfDarwin vs Lamarck Charles Darwin in 1854 Jean-Baptiste Lamarck

Mathematics in Life Sciences Proseminar Fall 2009 1/4/2010 BacterialFluctuationTest.doc Page 1

A Fluctuation Test in Bacteria: Using math to test Darwin’s theory

When exposed to an antibiotic, bacteria evolve resistance with astonishing speed. If you spread a culture of antibiotic-sensitive bacteria on a Petri plate containing an antibiotic, after overnight growth in an incubator a few colonies (clones) of bacterial survivors appear. These can be proved to be genetically resistant; they have DNA mutations that confer resistance. To many early microbiologists, this adaptive capacity of bacteria seemed too rapid to be explained by Darwinian processes. Microbiologist Salvador Luria called bacteriology “the last stronghold of Lamarckism.” In 1944, Luria and physicist-turned-molecular biologist Max Delbrück addressed the question of the origin of bacterial mutants with a beautiful experiment, the Luria-Delbrück fluctuation test. We will perform and analyze a version of the Luria-Delbrück experiment.

Darwin vs Lamarck

Charles Darwin in 1854 Jean-Baptiste Lamarck (1744-1829) Darwin's theory of evolution by natural selection differed from older theories of evolution, such as that of Lamarck, in the way it explained the origin of variation (mutation). Both Darwin and Lamarck believed that organisms adapted to their environment by acquiring favorable characteristics. However, Lamarck thought that these favorable traits were acquired in direct response to the environment, and then inherited by subsequent generations. Darwin proposed that mutations occurred by chance, before exposure to the selective environment. In the case of bacteria acquiring antibiotic resistance, Lamarck would have said that the drug caused the adaptive mutations to occur. Darwin would have said that the mutations had already occurred in a minority of bacteria within the populations, and were subsequently selected by the antibiotic, which killed the bacteria that weren’t resistant (the vast majority of the bacteria in the population) and allowed only the rare resistant mutants to grow and form colonies.


You can see that this theory might be tricky to test. Antibiotic resistant bacteria can only be observed by exposing bacteria to the antibiotic, so how can you tell if the mutations occurred before or after that exposure?1

Valine resistance in E coli K12 Valine, a branched-chain amino acid, is one of the 20 amino acids that make up proteins. Bacteria such as E. coli make their own valine and don’t require it to be added to their growth medium. Surprisingly, valine is toxic to E coli strain K12 when it’s cultured in medium without any other amino acids. In particular, when plated on GB/Val medium, a nutrient agar containing a high concentration of valine but no other amino acids, E coli K12 fails to form colonies. This poisoning can be overcome by supplementing the valine-containing medium with isoleucine, another branched chain amino acid that E coli normally makes for itself. The reason is that valine and isoleucine are both made from the same starting molecule (pyruvate) using the same enzymes. As an endproduct, valine regulates its biosynthesis by inhibiting the first enzyme in this pathway, but in excess valine also inhibits isoleucine biosynthesis. As a result, bacteria are starved for isoleucine, a condition that can be reversed by adding this amino acid. We will use valine as a selective agent in these experiments, just as we would an antibiotic. When E coli K12 bacteria are plated on GB/Val medium, although the vast majority of bacteria don’t survive (as explained above), a few surviving colonies are observed. These are valine-resistant mutants. Our version of the Luria-Delbrück experiment will ask if these mutants arose only after exposure to valine or if they pre-existed in the population before exposure to valine.

Materials for the experiment

E. coli K12 cells (just “K12” for short): stored in 100-µl portions in sterile tubes in a –80ºC freezer

GB medium: a sterile liquid medium containing: o Glucose (G) at high concentration to serve as carbon and energy source o Vitamin B1 (B) in trace amounts because K12 is unable to make its own

B1 o Mineral salts supplying all the minerals required for growth of K12 o No amino acids

GB/Val plates: petri dishes with sterile GB medium supplemented with a high concentration of valine and agar to solidify the liquid; only valine-resistant K12 mutants can form colonies on these plates

Rich medium plates: petri dishes with a sterile medium containing hydrolyzed milk protein and extract of yeast cells supplemented with agar to solidify the

1 Lest you think this is a dusty, antique controversy it was revived in 1988 by renowned geneticist John Cairns, who claimed to have observed mutations in bacteria which were directed by the growth environment. Cairns’s paper generated lively controversy and a fruitful line of research which uncovered new genetic mechanisms. By 2001, the Cairns controversy had been resolved in favor of Darwin, but with an enriched understanding of bacterial adaptation to stress.


liquid; this medium contains all the amino acids (as well as other nutrients), and allows K12 cells to grow whether or not they are valine resistant

Sterile culture vessels: o 1 × 3.5 inch culture tubes for 1–5 ml liquid cultures o 1-liter glass culture flasks for 100–200 ml liquid cultures o sterile disposable plastic 14-ml “snap-cap” tubes for 0.2-ml liquid cultures

Other sterile vessels (not used for culturing) o 50-ml plastic screw-cap tubes (just called “50-ml tubes”) o 60-ml plastic bottles o 50-ml bottles o 1.5-ml plastic microtubes with attached caps o 500-µl plastic microtubes with attached caps

Blue sterile L-shaped spreaders individually wrapped in paper/plastic sleeves 200-µl and 1000-µl pipetters, along with sterile yellow and blue tips; we’ll show

you how to use these in the lab vortexer: a device for thoroughly mixing the contents of a small tube; we’ll show

you how to use these in the lab Spectrophotometer; we’ll show you how to use this in the lab Non-sterile 13 × 100 mm glass tubes, which serve as the sample containers for

measurements with the spectrophotometer

Preparation for the experiment (done by the staff in the days before the proseminar)

1. Frozen K12 cells (Materials) were used to inoculate 2 ml GB medium in a 1 × 3.5 inch culture tube; the culture was shaken vigorously for ~24 hr at 37ºC; during this shaking, the original cells in the inoculum doubled many times until they became so crowded no more cell division could occur; this state of a culture is called the “stationary phase.” 2. Into two sterile 1.5-ml plastic microtubes labeled A and B pipette 495 µl GB medium; into tube A pipette 5 µl of the stationary phase culture previous step; close the cap of tube A and vortex to thoroughly mix the cells in with the rest of the liquid (this tube now contains a 1/100 dilution of the stationary-phase culture); now pipette 5 µl from tube A (just mixed) into tube B; close the cap of tube B and vortex (what dilution of the stationary phase culture does tube B contain?) 3. Into a bottle containing 200 ml sterile GB medium pipette 100 µl from tube B previous step; close the cap of the bottle and invert it many times to thoroughly mix the cells in with all the medium (what dilution of the stationary-phase culture does the bottle now contain?). 4. Pipette 200-µl portions of the inoculated medium in the bottle previous step onto 14 rich medium plates; use sterile blue spreaders (Materials) to spread the 200-µl puddles evenly over the plates; incubate the plates overnight in the 37º incubator; next day (after 14 hours of incubation) store the plates in the refrigerator so scholars can count colonies during the proseminar.


5. Pipette 200-µl portions of the inoculated medium in the bottle step 3 into 42 sterile 14-ml snap-cap tubes; cap the tubes loosely, so that air can pass into the tube but bacteria are unlikely to be able to enter from the air. 6. Pour the remaining inoculated medium in the bottle step 3 (theoretically ~186 ml remaining) into a sterile 1-liter culture flask. 7. Shake the flask (previous step) and the 42 snap-cap tubes (step 5) at 37º for ~2 days; this is enough time for all 43 cultures (the ~186-ml culture in the flask; the 42 200-µl cultures in the snap-cap tubes) to reach stationary phase. 8. Earlier in the week, a stationary phase culture like the one in the 1-liter culture flask step 7 was prepared; the cell concentration was measured at 4.97 × 109 cells/ml (for the time being we won’t explain how the cell concentration was determined); 30 ml of the culture were mixed with 30 ml GB medium to make a 1/2 (2-fold) dilution of the culture; various further dilutions of this 1/2 dilution were made in 13 × 100 mm glass tubes by mixing various volumes of the stationary phase culture with various volumes of GB medium (Materials) as specified in the table below; these dilutions were analyzed with the spectrophotometer (as you’ll learn how in the lab), with the results shown in the table.


Volume of GB medium (ml)

Volume of 1/2 culture dilution (ml)

Cells/ml in the dilution

Percent transmission at 600 nm

3.75 0 0 100 3.6 0.15 90.7 3.45 0.3 82.0 3.3 0.45 73.2 3.15 0.6 66.2 3 0.75 60.0 2.85 0.9 55.9 2.7 1.05 50.8 2.55 1.2 47.8 2.4 1.35 42.5 2.25 1.5 40.0 2.1 1.65 37.5 1.95 1.8 34.8 1.8 1.95 33.0 1.65 2.1 30.1 1.5 2.25 27.9 1.35 2.4 26.9 1.2 2.55 24.8 1.05 2.7 22.0 0.9 2.85 20.8 0.75 3 20.0 0.6 3.15 18.9 0.45 3.3 17.8 0.3 3.45 17.0 0.15 3.6 16.0 0 3.75 15.1

Fill in the third column of the table from the information given in this step; you’ll use these results as a calibration standard to determine the cell density in the stationary phase culture step 7 spectrophotometrically. 9. Just before the proseminar:

pour ~50 ml of the ~186-ml culture in the 1-liter culture flask step 7 into six sterile 60-ml bottles (one bottle for each team of MLS scholars)

transfer the 0.2-ml culture in each of the 42 snap-cap tubes step 7 into a sterile 500-µl microtube (Materials); close caps to preserve sterility


Instructions for Lab I This is what you’ll do in the lab (125 Life Sciences Center) during the proseminar period

(6:00 to ~7:00 pm, Monday September 14, 2009) NOTE: You’ll break into six teams of four scholars; it doesn’t matter if a team is short a person. Materials supplied 10. Each team will be supplied with:

seven GB/Val plates labeled “0.2-ml culture” on white labels with a red vertical stripe

seven GB/Val plates labeled “Bulk culture” on white labels with a blue vertical stripe

seven 500-µl microtubes with the 0.2-ml cultures step 9 in a small rack; labeled “0.2-ml culture” on white labels with red vertical stripe

a 60-ml bottle with ~50 ml of the bulk culture step 9 in a small rack; labeled “bulk culture” on white label with blue vertical stripe

20 individually wrapped sterile blue L-shaped spreaders (Materials; only 14 should be needed)

a large plastic beaker for discarding used contaminated materials (e.g., used microtubes, used spreaders, used pipette tips, etc.)

a 15-ml bottle of GB medium two glass 13 × 100 tubes in a small rack, for use in the spectrophotometer a box of yellow sterile 200-µl pipette tips a box of blue sterile 1000-µl pipette tips an adjustable 200-µl pipetter an adjustable 1000-µl pipetter a vortexer two or three petri plates from step 4 four Sharpie marking pens (one for each scholar)

Spreading the bulk culture and the 0.2-ml cultures on petri plates 11. Dial the 200-µl pipetter to 200 µl (its maximum capacity); put a sterile tip on the pipetter as instructed by the staff; swirl or invert the 60-ml bottle with the ~50 ml of bulk culture (see step 10) several times to ensure that the cells are evenly distributed in the medium; use the pipetter with its tip in place to sterilely pipette 200 µl of the bulk culture onto seven GB/Val plates labeled “Bulk culture” on white labels with a blue vertical stripe (see step 10); there is no need to change pipette tip between pipetting unless you have compromised its sterility in some way; discard the used pipette tips into the plastic biowaste beaker (see step 10); remove a blue cell spreader from its sleeve without compromising sterility; use it to spread the 200-µl puddle of culture as evenly as possible over the surface of each petri dish; use a different blue spreader for each petri dish; discard used spreaders into the plastic biowaste beaker; allow the petri plates to dry


right-side up on the bench2, so that the 200 µl of fluid is absorbed into the agar gel and cells are consequently stuck at their current location on the agar surface. After about an hour, staff will incubate the petri plates upside down in a 37º incubator for 48 hr, then store them in the refrigerator for you to analyze in the next proseminar. 12. Use the technique in the previous step to spread the entire contents of each 500-µl microtube (with one of the 0.2-ml cultures; labeled “0.2-ml culture” on a white label with a red vertical stripe) on one of the GB/Val petri plates labeled in the same way as the microtube. It’s mandatory that you use a different pipette tip and a different blue spreader for each of these petri plates!! Determining cell concentration by counting colonies 13. Count the colonies on the petri plates from step 4 (200-µl portions of the inoculated culture). The best way to do this is to hold the petri plate without its lid and the agar surface facing away from you up to a light or against a black surface so that the colonies are easily visualized; then use the Sharpie marking pen to mark each colony on the plastic underside of the petri dish as you count it; record the colony count both in your records and on the lid of the petri dish. Here is a table of your counts:

67 45 49 53 50 54 56 63 65 44 54 46 62 56

Average the 14 counts to determine a mean number of colonies per plate; on your own time, after proseminar, do the following two calculations (don’t give up!!! if you are having trouble doing these calculations, consult one of the other scholars or a faculty member): 2 Microbiologists usually incubate petri plates upside down so that liquid that condenses on the lid can’t drip down onto the agar surface. That is what we’ll do, too. However, because you’re spreading a relatively large volume on the agar surface, inverting the plates immediately runs the risk of the not-yet absorbed liquid sliding down the agar surface and pooling at a corner or even running down off the surface altogether.


since each plate represents 200 µl of the just-inoculated culture at step 3, and since each colony represents a single viable cells deposited on the petri plate, the mean colony count will allow you to determine the cell titer of the just-inoculated culture: the number of viable cells per ml.

Since the just-inoculated culture at step 3 is at the same time a dilution (see step 2) of the stationary phase culture (step 1) that was the ultimate source of the inoculum, the mean colony count will also allow you to determine the cell titer of the stationary phase from step 1

Determining cell concentration by spectrophotometry 14. Use the Sharpie marking pen to label two glass 13 × 100 mm tubes (step 10) Blank and Culture; into tube Blank pipette 3.5 ml GB medium (step 10; in a 15-ml plastic bottle; since the largest pipetter you have has a capacity of only 1 ml, you’ll have to do this by pipetting four 875-µl portions of GB into the glass tube3); in tube Culture make 3.5 ml of a 1/10 dilution of the bulk culture by adding the following volumes:

350 µl of the bulk culture (step 10; in the 60-ml bottle) 3.15 ml of GB medium (fill in the blank; again you’ll have to pipette four

portions: this time, four 787.5-µl portions4) If you compare the two tubes, you’ll see that tube Blank is clear while tube Culture is turbid (cloudy). Tube Culture is turbid because the bacteria suspended in the medium are scattering light. If a light beam is shone through tube Blank, it passes through the empty medium unimpeded and emerges at almost full strength through the other side of the tube. We’ll define the intensity of light that emerges at the other side of tube Blank as 100% transmission. In contrast, if the same light beam is shone through tube Culture, much of the light is lost to scattering in all different directions, and not all the light makes it all the way to the other side of the tube; the transmission through the tube is less than 100%. The higher the concentration of bacteria, the lower the percent transmission; percent transmission is thus an indirect measurement of bacterial cell concentration. By comparing the percent transmission for a sample with unknown cell concentration (the sample in your tube Culture) with the percent transmission from a graded series of samples with known cell concentrations (data at step 8), it’s possible to convert the percent transmission measurement to an estimate of actual cell concentration.

3 4 × 875 = 3500 µl 4 4 × 787.5 = 3150 µl


1 cm

lightbeam

photo-detector

shutter blockslight beam whentube removed

transmissionmeter

amplifiergain

dark currenttube

The spectrophotometer (see schematic diagram above) is designed to measure percent transmission of a light beam through samples in 13 × 100 mm glass tubes. In the diagram above, we’re looking downward into a tube that’s been inserted into the instrument’s sample well. In our experiment, the samples in the tube will be suspensions of bacteria in medium; the black specks are meant to represent the suspended bacteria. A beam of light with a single wavelength (we’ll use 600 nm) is shone through the tube from one side (from the left in the diagram), and the light that emerges at the other side of the tube (the right in the diagram) hits a photodetector, which produces an electrical signal that’s proportional to the intensity of the light beam (i.e., the number of photons hitting the photodetector per second). The electrical signal is amplified, and the amplified signal drives the transmission meter on the front of the instrument. Beside the wavelength adjustment, there are two controls, controlling the “dark current” and the amplifier’s gain. Here’s how the controls are used:

Dark current control: When there’s no tube in the sample well, a shutter falls into place, blocking the light beam so that no light reaches the photodetector. Theoretically the photodetector doesn’t send out any electrical signal in these circumstances, but in practice there’s a very small “dark current” even when there’s no light at all. The dark current control allows the user to offset this small dark current, so that no signal reaches the amplifier when there’s no light getting to the photodetector. How does the user accomplish this? Easy! Just turn the dark current knob (the one on the left-hand side on the front of the instrument) until the transmission meter reads 0% transmission (left-most tick on the black percent transmission scale). Don’t touch the dark current control once it has been properly adjusted.

Gain control: Now put the Blank tube (with empty GB medium) into the sample well. The shutter moves out of the way, so that the light that passes through the tube can reach the photodetector. Since there no bacteria in the blank tube to scatter light (i.e., the solution isn’t turbid), this is the maximum light that’s going to get through the tube—i.e., 100% transmission. Accordingly, adjust the gain control (the knob on the right-hand side of the front of the instrument) until the transmission meter reads 100% transmission (right-most tick on the black percent transmission scale). Don’t touch the gain control (or the dark current control) until you’ve made your measurement.


Measuring percent transmission: Remove the Blank from the sample well; vortex your Culture tube gently (if you vortex too vigorously, your sample will shoot out of the tube); insert the just-vortexed tube into the spectrophotometer; record the percent transmission on the transmission meter (NOTE: All six readings were in the vicinity of ~70%).

On your own time, use the transmission data in step 8 to construct a graph of percent transmission versus cell concentration (you’ll have to fill in the third column of the table in step 8); by comparing your measured percent transmission to that graph, estimate the concentration of cells in the undiluted bulk culture step 7. Don’t forget that the sample that you measured was a 1/10 dilution of the bulk culture! Later in the semester we’ll analyze how the spectrophotometer works more deeply when we take up Beer’s law; for the time being, though, we’ll just use the spectrophotometric transmission data without any deep understanding of them.

Contrasting Lamarckian and Darwinian accounts of mutation When each 0.2-ml culture is started from an initial inoculum of ~50 stationary-phase cells (see step 13 above), there will be a lag of a few hours while the cells adapt to the new culture conditions and resume cell division. After that lag, the cells will double every hour or so until the cell density reaches high enough levels to slow down cell division. About 36 hr after inoculation, when the cells finally reach stationary phase and cell division ceases altogether, there are ~109 cells in the culture. Altogether, it takes ~24 rounds of doubling for the cell number to increase from 50 to 109 (50 × 224 = 8.4 × 108, which is pretty close to 109). According to Lamarck, mutations to valine resistance happen only when induced by valine, meaning only when the cells are spread on the plate. According to Darwin, a mutation to valine resistance could happen at any time during culture growth preceding spreading. The consequences of the two contending theories are suggested in the following schematic diagram from Stan Maloy at San Diego State University (http://www.sci.sdsu.edu/~smaloy/MicrobialGenetics/topics/mutations/fluctuation.html):


LAMARCK DARWIN

The diagram shows four small cultures, in each of which there are three doublings before bacteria are spread on the plate containing valine5 (shading). (Of course, your actual cultures started with ~50 cells and reached ~109 cells after ~24 doublings.) The blackened ovals represent valine-resistant (ValR) mutants.

According to Lamarck, there are no such mutants before the cells are spread on the plate.

According to Darwin, new ValR mutants can appear any time on a random basis. If a ValR mutant happens to appear in an early cell division, there will be lots of mutants when the cells in that small culture are plated; such cultures are called “jackpot” cultures. If a ValR mutant doesn’t happen to appear at all, there will be no mutants on the plate.

5 Actually, Maloy was describing mutations that confer resistance to a phage (bacterial virus), not valine. But the principle of the fluctuation test doesn’t depend on the selective agent (valine vs phage), so we’ll pretend that he was talking about valine resistance.


Think about the consequences of the two theories for the expected dispersion of ValR colony numbers on the two types of plates. By “dispersion” we mean something like the scatteredness of the numbers. For example, in the scattergram below, the left-hand series (the diamonds) shows low dispersion6, while the right-hand series (the triangles) shows very high dispersion7. The variance of the numbers in a series is a quantitative measure of their dispersion, as we’ll detail later (and as you’ve already encountered in the Excel workshop).

0

50

100

150

200

250

Num

ber o

f col

onie

s

In light of the foregoing considerations, and in preparation for the next proseminar, try to answer the following two challenging questions:

Does the Lamarckian theory make different predictions about the dispersion of ValR colony numbers on your bulk-culture plates (42 200-µl samples from a single bulk culture) vs your 0.2-ml culture plates (42 separate 200-µl cultures grown under essentially the same conditions as the bulk culture)?

Does the Darwinian theory make different predictions about the dispersion of ValR colony numbers on your bulk-culture plates vs your 0.2-ml culture plates?

Summary of work to do before proseminar on Monday, September 21,

2009

Fill out the third column of the table at step 8. Use the data at step 8 plus your percent transmission data at step 14 to estimate

the cell concentration (cells/ml) in the bulk culture after growth (when the culture is in stationary phase).

Use the colony counts at step 13 to:

6 These are your actual colony count data from step 13. 7 These are made-up numbers whose mean is the same as the mean of the left-hand series.


o Estimate the cell concentration in the bulk culture immediately after inoculation at step 3 (before growth)

o Estimate the cell concentration in the 2-ml stationary-phase culture step 1 that was the source of the inoculum for the bulk culture at step 2.

Turn the above assignments as a single computer document attached as an e-mail to George Smith [email protected]. You won’t be graded on this work; we’re asking you to submit your answers as a way of monitoring how well we’re doing at teaching the concepts. In addition:

Try to answer the two questions in the previous subsection; it’s vital that you have considered them before the next proseminar on September 21. No need to turn in this assignment.

We recognize that these are challenging assignments. But you have help. You can use the e-mail functions of Blackboard (in the Communications content area) to e-mail each other to ask for help or ask questions to the all the scholars. You can e-mail George Smith or Miriam Golomb. You can call George Smith on his cell phone 999-1829.


Instructions for Lab II This is what you’ll do in the lab (125 Life Sciences Center) during the proseminar period

(6:00 to ~7:00 pm, Monday September 21, 2009) 15. Each team will be given seven 0.2-ml culture plates and seven bulk-culture plates from step 11; count them as you did the 14 plates at step 13. A scholar volunteer will enter your counts into a master Excel, which will be posted immediately on Blackboard by one of the staff. 16. Go to Blackboard and download the Excel file from the Documents content area; for each data series (bulk-culture and 0.2-ml culture) compute the mean (average) and variance of the colony counts, using the built-in Excel functions:

The mean (average) x of a series of N numbers },,...,,{ 121 NN xxxx − (the 42 bulk-culture or 42 0.2-ml culture colony counts in this case) is given by the obvious

formula N

x

Nxxxxx

N

ii

NN∑=− =

++++= 1121 ... . Excel has a convenient built-in

function for calculating the mean: just enter =average(###) into a cell, where ### is the range of cells you want to average (42 cells in this case). You can specify the range of cells simply by dragging the cursor through them.

The variance of a series of numbers },,...,,{ 121 NN xxxx − is the average of the squares of their deviations from their mean; it is given by the formula

( )( )

221

2

var xxN

xxx

N

ii

i −=−

=∑= . The second part of the equation says that the

variance is equal to the average of the squares of the numbers minus the square of the average of the numbers. It shouldn’t be too hard for you to prove that second part. Luckily, though, no elaborate calculations will be necessary in practice, since Excel has a convenient built-in function for calculating variance: just enter =var(###) into a cell, where again ### is the range of cells you want to average.

Once the means and variances are available, we’ll discuss the key questions (see above under Contrasting Lamarckian and Darwinian accounts of mutation) in the seminar room 121:

Does the Lamarckian theory make different predictions about the variance of ValR colony numbers on your bulk-culture plates (42 200-µl samples from a single bulk culture) vs your 0.2-ml culture plates (42 separate 200-µl cultures grown under essentially the same conditions as the bulk culture)?

Does the Darwinian theory make different predictions about the variance of ValR colony numbers on your bulk-culture plates vs your 0.2-ml culture plates?


The data from the scholars’ counts of their plates, the staff’s counts of the same plates, and the counts from the pilot experiment are gathered in the Excel file FluctuationTestData.xls, which is also posted in the Documents content area. The data are graphed in the scattergrams below:

Num

ber o

f col

onie

s

Bulkculture

0.2-mlcultures

Mean 31.6 12.5

Variance 64.6 2103.4

28.8 26.3

16.2 385.3

Bulkculture

0.2-mlcultures

Main experiment(staff count)

Num

ber o

f col

onie

s

0

10

20

30

4050

60

70

8090

100

Pilotexperiment

Which of the two mutation theories, Darwinian or Lamarckian, do these data support? This question will be revisited later in the proseminar, when the data can be analyzed more quantitatively once we learn about the binomial and Poisson distributions.

E S P

E l e c t r o n i c S c h o l a r l y P u b l i s h i n g

h t t p : / / w w w . e s p . o r g

MUTATIONS OF BACTERIA

FROM VIRUS SENSITIVITY TO VIRUS

RESISTANCE

S. E. LURIA AND M. DELBRÜCK

Luria, S. E., and M. Delbrück, 1943. Mutations of bacteria from virus sensitivity to virus resistance. Genetics, 28: 491–511.

New material in this electronic edition is

© 2001, Electronic Scholarly Publishing Project

http://www.esp.org

The original work, upon which this electronic edition is based, is © 1943, Genetics Society of America and is reprinted here with permission.

This electronic edition may be used for educational or scholarly purposes, provided that these copyright notices are included. The

manuscript may not be reprinted or redistributed, in any form (printed or electronic), for commercial purposes without written

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Electronic Scholarly Publishing Project Foundations Series: Classical Genetics

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Bibliographical Note

This ESP publication is a newly typeset, unabridged version, based on the original 1943 paper published by the Genetics Society of America in the journal Genetics. All footnotes are as they appeared in the original work. Figures 1 and 2 have been redrawn for this digital version.

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iii

INTRODUCTION

acteria have been the subject of intensive investigation ever since they were recognized as causative agents in disease. Despite this

scrutiny, for several reasons bacteria were regarded as unsuitable for genetic research. First, bacterial reproduction seemed to be totally asexual and thus could not be analyzed with standard genetic methods. Second, because no bacterial chromosomes could be detected under the light microscope and because nothing akin to the mitotic or meiotic segregation of chromosomes could ever be observed, many bacteriologists felt that bacteria simply could not possess genes similar to those of higher organisms. (Some even performed mathematical calculations to “prove” that the invisibly small bacterial “nucleus” was not big enough to hold genes.) Finally, bacteria exhibited patterns of inheritance that seemed to be fundamentally different from those of higher organisms. In particular, bacteria appeared to transmit acquired characteristics to their progeny. For example, whenever about a billion bacteria are mixed with a particular toxin, nearly all of the bacteria are killed. However, a few will survive and give rise to colonies that are permanently and specifically resistant to that and only to that particular toxin. Because such findings are routine and can be easily replicated with the same or with different treatments, many workers assumed that contact with the particular toxin somehow induced a few bacteria to acquire an inherited resistance which they then transmitted to their progeny. Since Lamarckian inheritance (the transmission of acquired characteristics) is not compatible with Mendelian mechanisms, and since a detailed non–Mendelian explanation for bacterial inheritance had in fact been proposed (the influential British physical chemist, Sir Cyril Hinshelwood, had offered mathematical models which he claimed proved that all instances of inherited variation in bacteria were due to induced changes in chemical equilibria, not genes), many researchers concluded that bacteria simply did not possess a genetic apparatus in the Mendelian sense. Other workers, however, were convinced that the inheritance systems of bacteria and higher organisms had to be fundamentally similar and that the apparent differences were illusions produced by inadequate experimental methods and by a conceptual failure to recognize the inheritance patterns produced by Mendelian mechanisms acting in vast populations of rapidly propagating haploid individuals.

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iv

The range of opinions can be appreciated from the following quotations.

Bacteria ... appear to be not only wholly asexual but premitotic. Their hereditary constitution is not differentiated into specialized parts with different functions. They have no genes in the sense of accurately quantized portions of hereditary substance; and therefore they have no need for the accurate division of the genetic system which is accomplished by mitosis. ... That occasional ‘mutations’ occur we know, but there is no ground for supposing that they are similar in nature to those of higher organisms, nor, since they are usually reversible according to conditions, that they play the same part in evolution. We must, in fact, expect that the processes of variation, heredity, and evolution in bacteria are quite different from the corresponding processes in multicellular organisms.

J. Huxley, 1943, Evolution: The Modern Synthesis, New York: Harper & Brothers Publishers, pp. 131–132

Most of these [bacterial variations] can be simply accounted for on the assumption that a variety of [gene] mutations arise, each with a certain frequency, in bacterial strains. In any given environment a certain biotype or biotypes are selected to become the dominant components of the culture. ... Since [genetic] mutation is usually reversible, the bacterial [variations] are likewise reversible in most cases. Although some bacteriologists are prone to believe that the behavior of bacteria is incompatible with established concepts of genetics and evolution theory, there are valid reasons to think that bacteria may prove to be the best available materials for exact studies on mutation and natural selection.

T. Dobzhansky, 1941, Genetics and the Origin of Species, New York: Columbia University Press, pp. 189–190

The wide disparity of opinions caused one researcher to observe in frustration, “The subject of bacterial variation and heredity has reached an almost hopeless state of confusion. Almost every possible view has been set forth and there seems no reason to hope that any uniform consensus of opinion may be reached in the near future.” Such pessimism proved unfounded. In 1943, Salvador E. Luria and Max Delbrück showed that apparent examples of Lamarckian inheritance were actually due to true genetic mutation, and in 1946 Edward Tatum and Joshua Lederberg showed that both linkage and recombination could be detected in bacteria. Immediately after these breakthrough discoveries, many researchers were attracted to microbial genetics and soon research on these “unsuitable” organisms was providing the foundation upon which the new edifice of molecular biology was to be built.

v

Demonstration of True Genetic Mutation in Bacteria

The first demonstration of true genetic mutation in bacteria involved a study of the interaction between one wild–type strain of bacteria (E. coli B) and a particular kind of virus that attacks, multiplies within, and then destroys the bacteria, liberating hundreds of progeny phage in the process. Since we will be discussing bacterial viruses (or bacteriophage, or even just phage, as they are sometimes known) at length in the following chapter, here we will merely provide a brief description so that you may appreciate their role in this experiment. Bacteriophage are so small that they are totally invisible, even to the best light microscope. However, even before the advent of the electron microscope, many of their attributes could be determined through an analysis of their effects. For example, they were known to occur as a variety of true–breeding types with precise differences in the strains of bacteria they could attack. True–breeding types of phage are identified with specific symbolic designations, such as T1, T2, T3, and so on. Almost always, when an excess of T1 phage is mixed with about 105 of E. coli B and then plated, no bacterial colonies appear. Every bacterium is infected and killed by phage. However, if 109 bacteria are used, the chances are good that a few colonies will appear, indicating that some rare bacteria either possess or are capable of acquiring resistance to T1 phage. Furthermore, if these surviving colonies are used to establish pure cultures, it is found that all of the bacteria in these cultures exhibit T1 resistance. This resistance is usually quite specific and does not affect the sensitivity of the bacteria to other phage, such as T2 through T7. Initially, such observations led to an intense debate regarding the origin of the T1–resistant cells. The key dispute was whether or not any resistant cells existed in the culture prior to the application of phage. Supporters of the MUTATION THEORY felt that resistance was due to random genetic mutation and that the resistant bacteria were present before the treatment. Adherents to the ADAPTATION THEORY asserted that the resistance was too precisely adapted to one phage to be explained as the result of random mutation. Thus, they argued that resistance occurred only as a specific physiological adaptation to contact with a particular phage and, therefore, that the resistant bacteria were not present prior to treatment. Since these two hypotheses made unambiguously different claims regarding the existence of resistant bacteria prior to treatment with phage, you might think that a simple test for the presence of such bacteria could have demonstrated which was correct. At the time, however, resistant bacteria could be identified only by the application

vi

of phage and thus a direct test was impossible. Consequently, researchers struggled to find a valid indirect test. The first success was reported by Salvador E. Luria and Max Delbrück. Although their actual analysis involved some moderately sophisticated mathematical theory, we can summarize their approach as follows. First, let us recall that bacteria usually reproduce through simple division and let us assume that resistant bacteria are in fact produced by true genetic mutation. The accumulation, over time, of resistant mutants in a growing population could then be illustrated as follows, with the shaded circles representing resistant bacteria and with the asterisks denoting individuals in which a new mutation first occurred.

**

If resistance is due to heritable mutation, then two conclusions follow: (1) the proportion of resistant bacteria in a culture would increase over time (since existing mutants continue to reproduce while new mutants are continuously added), and (2) resistant bacteria would occur as groups of closely related individuals (since each new mutant will give rise to a related cluster of resistant progeny). On the other hand, if resistance is produced by physiological adaptation, then neither of these claims would be true. According to the adaptation theory, every bacterium has a small, but constant and independent probability of developing resistance upon contact with phage and this leads to different predictions: (1) the proportion of resistant bacteria in a culture would be constant over time (since all bacteria have the same constant probability of becoming resistant), and (2) resistant bacteria would occur as separate and scattered individuals (since it is assumed that every acquisition of resistance is an independent event with no genetic component). Two experimental tests to resolve the hypotheses are possible: (1) cultures could be examined to see if the proportion of resistants does increase over time, and (2) groups of related bacteria could be examined to see if the occurrence of resistance is correlated with genetic descent. Luria first attempted to measure the proportion of

vii

resistant bacteria as a function of culture age, but upon discovering that these measures varied unreliably from culture to culture he decided instead to see if resistance did in fact run in family groups.1 How could this be accomplished? Luria was dealing with liquid cultures containing billions of bacteria. By what possible means could he demonstrate that different resistant bacteria were in fact related to each other? After struggling with the problem for months, Luria finally realized that by applying probability theory to the variability that had been causing his problems, he could devise the necessary test. (Luria’s earlier training in physics presumably facilitated his ability to perceive the quantitative implications in his problem. Recall that Mendel’s prior training in probability theory made it possible for him to devise a genetic explanation for the numerical data he obtained from his pea plants.) To understand the logic behind Luria’s analysis, let us return to the pyramid of bacterial descendants illustrated above. The final generation contains a total of 64 bacteria, twelve of which are resistants that occur in two families, or CLONES (a “clone” is a population of cells all of which are asexual descendants from a single ancestor). Now, suppose that these 64 bacteria are thoroughly mixed and then subdivided at random into eight groups of eight bacteria each. Next, assume that each of these eight groups is tested and scored for the number of resistant bacteria it contains and that finally the series of measurements (i.e., the numbers of resistants per test group) is analyzed statistically. Such a statistical analysis requires at least two steps to determine first the central tendency and then the dispersion of the values. The central tendency is examined by calculating the mean number of resistants per group. The dispersion, or variability, of the results is determined by calculating the average distance between each value and the mean. Statisticians have shown (through theoretical analysis beyond the scope of this book) that the best distance measurement is the squared difference between each particular value and the mean. (For example, if the mean of a population of values is 17 and if a particular value is 22, then the squared difference between that value and the

1 The prediction that the proportion of resistants will constantly increase

over time is really true only for cultures exhibiting exponential growth. With the techniques available at the time, Luria could not maintain cultures in exponential growth long enough to obtain reliable measurements taken from the same colony at different times. However, once the chemostat –– a device that can maintain a population of bacteria in exponential growth indefinitely –– was invented in 1949, this prediction could be tested. The results obtained confirmed those obtained by Luria in his analysis of family groups.

viii

mean is (22–17)2, or 25. In formal statistics, the average of all the squared differences between each separate value and the mean is known as the variance of the population. The following table gives four different possible outcomes that might reasonably occur following the random distribution of 12 resistant and 52 sensitive bacteria into eight groups of eight bacteria each. Each row in the table gives the number of resistant bacteria in each of the eight groups for one possible outcome. The means and variances for each possible outcome are also shown.

group number 1 2 3 4 5 6 7 8 mean variance

3 2 3 1 0 1 0 2 1.50 1.25 2 4 0 2 0 1 1 2 1.50 1.50 1 3 0 2 0 1 4 1 1.50 1.75 2 3 0 0 0 3 2 2 1.50 1.50

The central tendency (as measured by the mean) must be 1.5 for all groups, because in every possible outcome the twelve resistant bacteria are always distributed into the eight groups, and twelve divided by eight is 1.5. Notice, however, that the variance can take on different values, depending upon how evenly the random process distributes the resistant bacteria to the different groups. Notice also that although the variance differs from one population to the next, it does not differ by much. This is because purely random distribution processes tend to produce fairly stable and predictable dispersions and thus fairly stable and predictable variances. In fact, if the particular random process underlying a given experimental situation is sufficiently understood, the governing theoretical probability distribution may be determined and used to calculate the expected mean and the expected variance for any series of measurements on results generated by that process. The Poisson probability distribution is the appropriate distribution to apply in situations such as the random distribution of rare, resistant bacteria into test groups. One of the recognizable attributes of a Poisson–generated population of values is that the population possesses a variance approximately equal to its mean. Thus, if resistant bacteria are in fact distributed randomly into test groups, measurements on those test groups should produce values that are consistent with a Poisson distribution –– specifically, the mean and the variance should be approximately equal. A comparison of the means and variances in the

ix

table above shows that they generally exhibit the expected relationship, and this is reasonable since the values were in fact generated randomly. Now, however, let us consider a situation in which the bacteria are distributed by a nonrandom process to the test groups. Specifically, let us suppose that the bacteria are assorted according to relatedness (i.e., by clones descended from ancestors at the generation marked with the arrows) and then tested in groups of eight, as below.

0 8 0 4 0 0 0 0

**

Here, the mean number of mutants per group is again 1.5, but the variance is now 7.75. This is much greater than expected under the assumption of random distribution. Grouping the bacteria into clones has resulted in all of the mutants being contained in two “jackpot” test groups, while the other six groups have none. If the mutation hypothesis is correct, then testing in clones should yield non–Poisson results. On the other hand, the adaptation hypothesis asserts that each resistant bacterium occurs as a separate, random event that is wholly unaffected by genetic relatedness. Therefore, if the adaptation hypothesis is correct, then testing in clones should yield Poisson results. These predictions can be summarized as follows, where the entries in the table give the predicted results for each hypothesis under each test condition.

adaptation

mutation

tested inclonal groups

tested afterrandom mixing

Poissondistribution

Poissondistribution

Poissondistribution

non-Poissondistribution

x

An indirect, but precise experimental test of the hypotheses is possible. All that has to be done is to compare the distribution of resistant bacteria in randomly mixed populations with the distribution in clonally derived populations. If both sets of populations show a Poisson distribution of resistants, then the adaptation hypothesis is correct. On the other hand, if the clonally derived populations show a non–Poisson distribution, then the mutation hypothesis is correct. With this insight,2 Luria went immediately to the laboratory to set up an experiment. Because the crucial part of the experiment involved measuring whether or not the group–to–group fluctuation in numbers was the same or greater than that expected to be produced by chance alone, the method became known as the FLUCTUATION TEST. Forty–eight hours later Luria had his results: a Poisson distribution (variance equal to the mean) occurred whenever the bacteria were tested after random mixing, but a decidedly non–Poisson distribution (many jackpots and many zeros, and a variance more than one hundred times greater than the mean) was observed whenever the bacteria were tested as clones. Elated that his data apparently disproved the adaptation theory, but worried that perhaps there was a flaw in his reasoning, he wrote to Max Delbrück (with whom he had been collaborating on phage research) and explained the experiment and results. The reply, on a post card, read, “I believe you have something important. I am working out the mathematical theory.” Delbrück’s assessment was too cautious. In fact, the fluctuation test must be regarded as the founding of bacterial genetics since it gave the first real proof that bacteria both possessed genes and experienced mutation. Luria and Delbrück shared the 1969 Nobel Prize with Alfred Hershey. Luria and Delbrück were also able to use their data to calculate the actual mutation rate per bacterial cell division. Averaged across all of 2 If you suspect that such experimental designs are conceived while the

researcher is poring over dry mathematical texts, think again. In his autobiography Luria explained how the idea behind the fluctuation test came to him: "I struggled with the problem for several months, mostly in my own thoughts, and also tried a variety of experiments, none of which worked. The answer finally came to me in February 1943 in the improbable setting of a faculty dance at Indiana University. ... During a pause in the music I found myself standing near a slot machine, watching a colleague putting dimes into it. Though losing most of the time, he occasionally got a return. Not a gambler myself, I was teasing him about his inevitable losses, when he suddenly hit the jackpot, ..., gave me a dirty look, and walked away. Right then I began giving some thought to the actual numerology of slot machines; in so doing it dawned on me that slot machines and bacterial mutations have something to teach each other."

xi

their experiments, this came to approximately 2.45 x 10–8. Thus, they not only proved that true genetic mutations occurred in bacteria, but also that such mutations were just as rare in bacteria as they were in higher organisms. Their work demonstrated that heritable variation in bacteria could be attributed to mechanisms similar to those in higher organisms. The previously puzzling ability of bacteria to respond rapidly and adaptively to changes in the environment could now be recognized as nothing more than the normal consequence of random gene mutation, followed by selection, in huge, rapidly reproducing populations. Following this discovery, many researchers hurried to determine the range of true genetic mutation occurring in bacteria. Soon, such variation was detected in virtually every trait that could be studied, such as color, colony morphology, virulence (ability to infect a host), resistance to antimicrobial agents, nutritional requirements, and fermentation abilities (i.e., the ability to use different compounds as carbon sources). Robert J. Robbins Seattle, Washington 2001

New material in this electronic edition is

© 2001, Electronic Scholarly Publishing Project

http://www.esp.org

The original work, upon which this electronic edition is based, is © 1943, Genetics Society of America and is reprinted here with permission.

Luria, S. E., and M. Delbrück, 1943. Mutations of bacteria from virus sensitivity to virus resistance. Genetics, 28: 491–511.

MUTATIONS OF BACTERIA FROM VIRUS

SENSITIVITY TO VIRUS RESISTANCE3,4

S. E. LURIA5 AND M. DELBRÜCK

Indiana University, Bloomington, Indiana, and Vanderbilt University, Nashville, Tennessee

INTRODUCTION

WHEN A PURE BACTERIAL CULTURE IS ATTACKED by a bacterial virus, the culture will clear after a few hours due to destruction of the sensitive cells by the virus. However, after further incubation for a few hours, or sometimes days, the culture will often become turbid again, due to the growth of a bacterial variant which is resistant to the action of the virus. This variant can be isolated and freed from the virus and will in many cases retain its resistance to the action of the virus even if subcultured through many generations in the absence of the virus. While the sensitive strain adsorbed the virus readily, the resistant variant will generally not show any affinity to it. The resistant bacterial variants appear readily in cultures grown from a single cell. They were, therefore, certainly not present when the culture was started. Their resistance is generally rather specific. It does not extend to viruses that are found to differ by other criteria from the

3 Theory by M. D., experiments by S. E. L. 4 Aided by grants from the Dazian Foundation for Medical Research and

from the Rockefeller Foundation. 5 Fellow of the Guggenheim Foundation.

2 S. E. LURIA AND M. DELBRÜCK (1943)

ESP Foundat ions Ser ies

strain in whose presence the resistant culture developed. The variant may differ from the original strain in morphological or metabolic characteristics, or in serological type or in colony type. Most often, however, no such correlated changes are apparent, and the variant may be distinguished from the original strain only by its resistance to the inciting strain of virus. The nature of these variants and the manner in which they originate have been discussed by many authors, and numerous attempts have been made to correlate the phenomenon with other instances of bacterial variation. The net effect of the addition of virus consists of the appearance of a variant strain, characterized by a new stable character — namely, resistance to the inciting virus. The situation has often been expressed by saying that bacterial viruses are powerful “dissociating agents.” While this expression summarizes adequately the net effect, it must not be taken to imply anything about the mechanism by which the result is brought about. A moment’s reflection will show that there are greatly differing mechanisms which might produce the same end result. D’HERELLE (1926) and many other investigators believed that the virus by direct action induced the resistant variants. GRATIA (1921), BURNET (1929), and others, on the other hand, believed that the resistant bacterial variants are produced by mutation in the culture prior to the addition of virus. The virus merely brings the variants into prominence by eliminating all sensitive bacteria. Neither of these views seems to have been rigorously proved in any single instance. BURNET’S (1929) work on isolations of colonies, morphologically distinguishable prior to the addition of virus, which proved resistant to the virus comes nearest to this goal. His results appear to support the mutation hypothesis for colony variants. It may seem peculiar that this simple and important question should not have been settled long ago, but a close analysis of the problem in hand will show that a decision can only be reached by a more subtle quantitative study than has hitherto been applied in this field of research. Let us begin by restating the basic experimental finding. A bacterial culture is grown from a single cell. At a certain moment the culture is plated with virus in excess. Upon incubation, one finds that a very small fraction of the bacteria survived the attack of the virus, as indicated by the development of a small number of resistant colonies, consisting of bacteria which do not even adsorb the virus. Let us focus our attention on the first generation of the resistant variant — that is, on those bacteria which survive immediately after the virus has been added. These survivors we may call the “original variants.” We know that these bacteria and their offspring are resistant

Mutations of bacteria from virus sensitivity to virus resistance 3

CL A S S I C A L GE N E T I C S

to the virus. We may formulate three alternative hypotheses regarding them. a. Hypothesis of mutation to immunity. The original variants were resistant before the virus was added, and, like their offspring, did not even adsorb it. On this hypothesis the virus did not interact at all with the original variants, the origin of which must be ascribed to “mutations” that occur quite independently of the virus. Naming such hereditary changes “mutations” of course does not imply a detailed similarity with any of the classes of mutations that have been analyzed in terms of genes for higher organisms. The similarity may be merely a formal one. b. Hypothesis of acquired immunity. The original variants interacted with the virus, but survived he attack. We may then inquire into the predisposing cause which effected the survival of these bacteria in contradistinction to the succumbing ones. The predisposing cause may be hereditary or random. Accordingly we arrive at two alternative hypotheses — namely,

b1. Hypothesis of acquired immunity of hereditarily predisposed indi-viduals. The original variants originated by mutations occurring independently of the presence of virus. When the virus is added, the variants will interact with it, but they will survive the interaction, just as there may be families which are hereditarily predisposed to survive an otherwise fatal virus infection. Since we know that the offspring of the original variants do not adsorb the virus, we must further assume that the infection caused this additional hereditary change.

b2. Hypothesis of acquired immunity — hereditary after injection. The original variants are predisposed to survival by random physiological variations in size, age, etc. of the bacteria, or maybe even by random variations in the point of attack of the virus on the bacterium. After survival of such random individuals, however, we must assume that their offspring are hereditarily immune, since they do not even adsorb the virus.

These alternative hypotheses may be grouped by first considering the origin of the hereditary difference. Do the original variants trace back to mutations which occur independently of the virus, such that these bacteria belong to a few clones, or do they represent a random sample of the entire bacterial population? The first alternative may then be subdivided further, according to whether the original variants do or do not interact with the virus. Disregarding for the moment this subdivision, we may formulate two hypotheses:

1. First hypothesis (mutation): There is a finite probability for any bacterium to mutate during its lifetime from “sensitive” to “resistant.” Every offspring of such a mutant will be resistant, unless reverse



mutation occurs. The term “resistant” means here that the bacterium will not be killed if exposed to virus, and the possibility of its interaction with virus is left open.

2. Second hypothesis (acquired hereditary immunity): There is a small finite probability for any bacterium to survive an attack by the virus. Survival of an infection confers immunity not only to the individual but also to its offspring. The probability of survival in the first instance does not run in clones. If we find that a bacterium survives an attack, we cannot from this information infer that close relatives of it, other than descendants, are likely to survive the attack.

The last statement contains the essential difference between the two hypotheses. On the mutation hypothesis, the mutation to resistance may occur any time prior to the addition of virus. The culture therefore will contain “clones of resistant bacteria” of various sizes, whereas on the hypothesis of acquired immunity the bacteria which survive an attack by the virus will be a random sample of the culture. For the discussion of the experimental possibility of distinction between these two hypotheses, it is important to keep in mind that the offspring of a tested bacterium which survives is resistant on either hypothesis. Repeated tests on a bacterium at different times, or on a bacterium and on its offspring, could therefore give no information of help in deciding the present issue. Thus, one has to resort to less direct methods. Two main differences may be derived from the hypotheses: First, if the individual cells of a very large number of microcolonies, each containing only a few bacteria, were examined for resistance, a pronounced correlation between the types found in a single colony would be expected on the mutation hypothesis, while a random distribution of resistants would be expected on the hypothesis of acquired hereditary immunity. This experiment, however, is not practicable, both on account of the difficulty of manipulation and on account of the small proportion of resistant bacteria. Second, on the hypothesis of resistance due to mutation, the proportion of resistant bacteria should increase with time, in a growing culture, as new mutants constantly add to their ranks. In contrast to this increase in the proportion of resistants on the mutation hypothesis, a constant proportion of resistants may be expected on the hypothesis of acquired hereditary immunity, as long as the physiological conditions of the culture do not change. To test this point, accurate determinations of the proportion of resistant bacteria in a growing culture and in successive sub-cultures are required. In the attempt to determine accurately the proportion of resistant bacteria, great variations of the proportions were found, and results did not seem to be reproducible from day to day.



Eventually, it was realized that these fluctuations were not due to any uncontrolled conditions of our experiments, but that, on the contrary, large fluctuations are a necessary consequence of the mutation hypothesis and that the quantitative study of the fluctuations may serve to test the hypothesis. The present paper will be concerned with the theoretical analysis of the probability distribution of the number of resistant bacteria to be expected on either hypothesis and with experiments from which this distribution may be inferred. While the theory is here applied to a very special case, it will be apparent that the problem is a general one, encountered in any case of mutation in uniparental populations. It is the belief of the authors that the quantitative study of bacterial variation, which until now has made such little progress, has been hampered by the apparent lack of reproducibility of results, which, as we shall show, lies in the very nature of the problem and is an essential element for its analysis. It is our hope that this study may encourage the resumption of quantitative work on other problems of bacterial variation.

THEORY

The aim of the theory is the analysis of the probability distributions of the number of resistant bacteria to be expected on the hypothesis of acquired immunity and on the hypothesis of mutation. The basic assumption of the hypothesis of acquired hereditary immunity is the assumption of a fixed small chance for each bacterium to survive an attack by the virus. In this case we may therefore expect a binomial distribution of the number of resistant bacteria, or, in cases where the chance of survival is small, a Poisson distribution. The basic assumption of the mutation hypothesis is the assumption of a fixed small chance per time unit for each bacterium to undergo a mutation to resistance. The assumption of a fixed chance per time unit is reasonable only for bacteria in an identical state. Actually the chance may vary in some manner during the life cycle of each bacterium and may also vary when the physiological conditions of the culture vary, particularly when growth slows down on account of crowding of the culture. With regard to the first of these variations, the assumed chance represents the average chance per time unit, averaged over the life cycle of a bacterium. With regard to the second variation, it seems reasonable to assume that the chance is proportional to the growth rate of the bacteria. We will then obtain the same results as on the simple



assumption of a fixed chance per time unit, if we agree to measure time in units of division cycles of the bacteria, or any proportional unit. We shall choose as time unit the average division time of the bacteria, divided by ln 2, so that the number Nt of bacteria in a growing culture as function of time t follows the equations (1) dNt/dt = Nt, and Nt = N0 et. We may then define the chance of mutation for each bacterium during the time element dt as (2) a dt, so that a is the chance of mutation per bacterium per time unit, or the “mutation rate.” If a bacterium is capable of different mutations, each of which results in resistance, the mutation rate here considered will be the sum of the mutation rates associated with each of the different mutations. The number dm of mutations which occur in a growing culture during a time interval dt is then equal to this chance (2) multiplied by the number of bacteria,6 or (3) dm = a dt Nt; and from this equation the number m of mutations which occur during any finite time interval may be found by integration to be (4) m = a (Nt - N0) or, in words, to be equal to the chance of mutation per bacterium per time unit multiplied by the increase in the number of bacteria. The bacteria which mutate during any time element dt form a random sample of the bacteria present at that time. For small mutation rates, their number will therefore be distributed according to Poisson’s law. Since the mutations occurring in different time intervals are quite independent from each other, the distribution of all mutations will also be according to Poisson’s law. This prediction cannot be verified directly, because what we observe, when we count the number of resistant bacteria in a culture, is not the number of mutations which have occurred, but the number of resistant bacteria which have arisen by multiplication of those which 6 We assume that the number of resistant bacteria is at all times small in

comparison with the total number of bacteria. If this condition is not fulfilled, the total number of bacteria in this equation has to be replaced by the number of sensitive bacteria. The subsequent theoretical developments will then become a little more complicated. For the case studied in the experimental part of this paper the condition is fulfilled.



mutated, the amount of multiplication depending on how far back the mutation occurred. If, however, the premise of the mutation hypothesis can be proved by other means, the prediction of a Poisson distribution of the number of mutations may be used to determine the mutation rate. It is only necessary to determine the fraction of cultures showing no mutation in a large series of similar cultures. This fraction p0, according to theory, should be: (5) p0 = e-m From this equation the average number m of mutations may be calculated, and hence the mutation rate a from equation (4). Let us now turn to the discussion of the distribution of the number of resistant bacteria. The average number of resistant bacteria is easily obtained by noting that this number increases on two accounts — namely, first on account of new mutations, second on account of the growth of resistant bacteria from previous mutations. During a time element dt the increase on the first account will be, by equation (3): a dt Nt. Nt, the number of bacteria present at time t, is given by equation (1). The increase on the second account will depend on the growth rate of the resistant bacteria. In the simple case, which we shall treat here, this growth rate is the same as that of the sensitive bacteria, and the increment on this account is ρ dt, where ρ is the average number of resistant bacteria present at time t. We have then as the total rate of increase of the average number of resistant bacteria dρ/dt = a Nt + ρ and upon integration (6) ρ = t a Nt if we assume that at time zero the culture contained no resistant bacteria. It will be seen that the average number of resistant bacteria increases more rapidly than the total number of bacteria. Indeed the fraction of resistant bacteria in the culture increases proportionally to time. This, as pointed out in the introduction, is a distinguishing feature of the mutation hypothesis but unfortunately, as will be seen in the sequel, is not susceptible to experimental verification due to statistical fluctuations. The resistant bacteria in any culture may be grouped, for the purpose of this analysis, into clones, taking together all those which derive from the same mutation. We may say that the culture contains clones of various age and size, calling “age” of a clone the time since its parent mutation occurred and “size” of a clone the number of bacteria in a clone at the time of observation. It is clear that size and



age of a clone determine each other. If, in particular, we make the simplifying hypothesis that the resistant bacteria grow as fast as the normal sensitive strain, the relation between size and age will be expressed by ‘equation (1), with appropriate meaning given to the symbols. The relation implies that the size of a clone increases exponentially with its age. On the other hand, the frequency with which clones of different ages may be encountered in any culture must decrease exponentially with age, according to equations (3) and (1). Combining these two results — namely, that clone size increases exponentially with clone age and that frequency of clones of different age decreases exponentially with clone age — we see that the two factors cancel when the average number of bacteria belonging to clones of one age group is considered In other words, at the time of observation we shall have, on the average, as many resistant bacteria stemming from mutations which occurred during the first generation after the culture was started as stemming from mutations which occurred during the last generation before observation, or during any other single generation. On the other hand, for small mutation rates it is very improbable that any mutation will occur during the early generations of a single or of a limited number of experimental cultures. It follows that the average number of resistant bacteria derived from a limited number of experimental cultures will, probably, be considerably smaller than the theoretical value given by equation (6), and, improbably, the experimental value will be much larger than the theoretical value. The situation is similar to the operation of a (fair) slot machine, where the average return from a limited number of plays is probably considerably less than the input, and improbably, when the jackpot is hit, the return is much bigger than the input. This result characterizes the distribution of the number of resistant bacteria as a distribution with a long and significant tail of rare cases of high numbers of resistant bacteria, and therefore as a distribution with an abnormally high variance. This variance will be calculated below. For such distributions the averages derived from limited numbers of samples yield very poor estimates of the true averages. Somewhat better estimates of the averages may in such cases be obtained by omitting, in the calculation of the theoretical averages, the contribution to these averages of those events which probably will not occur in any of our limited number of samples. We may do this, in the integration leading to equation (6), by putting the lower limit of integration not at time zero, when the cultures were started, but at a certain time t0, prior to which mutations were not likely to occur in any of our experimental



cultures. We then obtain as a likely average r of the number of resistant bacteria in a limited number of samples, instead of equation (6), (6a) r = (t - t0) a Nt. It now remains to choose an appropriate value for the time interval t – t0. For this purpose we return to equation (4), in which it was stated that the average number of mutations which occur in a culture is equal to the mutation rate multiplied by the increase of the number of bacteria. Let us then choose t0 such that up to that time just one mutation occurred, on the average, in a group of C similar cultures, or 1 = a C (Nt0 - N0). In this equation we may neglect N0, the number of bacteria in each inoculum, in comparison with Nt0, the number of bacteria in each culture at the critical time t0. We may also express Nt0 in terms of Nt, the number of bacteria at the time of observation, applying equation (1): Nt0 = Nt e-(t-t0). We thus obtain (7) t - t0 = ln (Nt Ca). Equations (6a) and (7) may be combined to eliminate t - t0 and to yield a relation between the observable quantities r and Nt on the one hand and the mutation rate a on the other hand, to be determined by this equation: (8) r = a Nt ln (Nt C a). This simple transcendental equation determining a may be solved by any standard numerical method. In figure 1, the relation between r and a Nt is plotted for several values of C. Estimates of a obtained from equation (8) will be too high if in any of the experimental cultures a mutation happened to occur prior to time t0. From the definition of t0 it will be seen that this can be expected to happen in little more than half of the cases. While we have thus obtained a relation permitting an estimate of the mutation rate from the observation of a limited number of cultures, this relation is in no way a test of the correctness of the underlying assumptions and, in particular, is not a test of the mutation hypothesis itself. In order to find such tests of the correctness of the assumption we must derive further quantitative relations concerning the distribution of



the number of resistant bacteria and compare them with experimental results.

1 10 102 103 aNt.1

1

10

102

.1

.1

.2

.5

1

2

5

10.1 .2 .5 1 2 5

103

104

105

C =100 C = 5 C = 1

C=1C=5

C=100

C=10C=20

r

Figure 1. — The value of a Nt as a function of r for various values of C. The upper left hand part of the figure gives the curves for low values of a Nt and of r on a larger scale. See text.

Since we have seen that the mutation hypothesis, in contrast to the hypothesis of acquired immunity, predicts a distribution of the number of resistant bacteria with a long tail of high numbers of resistant bacteria, the determination of the variance of the distribution should be helpful in differentiating between the two hypotheses. We may here again determine first the true variance — that is, the variance of the complete distribution — and second the likely variance in a limited number of cultures by omitting those cases which are not likely to occur in a limited number of cultures. The variance may be calculated in a simple manner by considering separately the variances of the partial distributions of resistant bacteria, each partial distribution comprising the resistant bacteria belonging to clones of one age group. The distribution of the total number of resistant bacteria is the resultant of the superposition of these independent partial distributions. Each partial distribution is due to the mutations which occurred during a certain time interval dτ, extending from (t - τ) to (t – τ + dτ).



The average number of mutations which occurred during this interval is, according to equation (3), (9) dm = a Nτ dτ = a Nτ e-τ dτ. These mutations will be distributed according to Poisson’s law, so that the variance of each of these distributions is equal to the mean of the distribution. We are however not interested in the distribution of the number of mutations but in the distribution of the number of resistant bacteria which stem from these mutations at the time of observation — that is, after the time interval τ. Each original mutant has then grown into a clone of size eτ. The distribution of the resistant bacteria stemming from mutations occurred in the time interval dτ has therefore an average value which is eτ times greater than the average number of mutations, and a variance which is e2τ times greater than the variance of the number of mutations. Thus we find for the average number of resistant bacteria: dρ = a Nt dτ, and for the variance of this number vardρ = a Nt eτ dτ. From this variance of the partial distribution, the variance of the distribution of all resistant bacteria may be found simply by integrating over the appropriate time interval — that is, either from time t to time 0 (τ from 0 to t), if the true variance is wanted, or from time t to time t0 (τ from 0 to t – t0), if the likely variance in a limited number of cultures is wanted. In the first case we obtain: (10) varρ = a Nt (et - 1). In the second case we obtain: (10a) varτ = a Nt [e(t-t0) - 1]. Substituting here the previously found value of (t- t0) and neglecting the second term in the brackets, we obtain: (11) varτ = C a2 Nt

2. Comparing this value of the likely variance with the value of the likely average, from equation (8), we see that the ratio of the standard deviation to the average is: (12) var lnt tr C N Ca/ / ( )= .



It is seen that this ratio depends on the logarithm of the mutation rate and will consequently be only a little smaller for mutation rates many thousand times greater than those considered in the experiments reported in this paper. In the beginning of this theoretical discussion we pointed out that the hypothesis of acquired immunity leads to the prediction of a distribution of the number of resistant bacteria according to Poisson’s law, and therefore to the prediction of a variance equal to the average. On the other hand, if we compare the average, equation (8), with the variance, equation (11), (not, as above, with the square root of the variance), we obtain (12a) varτ = r Nt C a / ln(Nt C a). Equation (12a) shows that the likely ratio between variance and average is much greater than unity on the hypothesis of mutation, if (Nt C a), the total number of mutations which occurred in our cultures, is large compared to unity.7 It is possible to carry the analysis still further and to evaluate the higher moments of the distribution function of the number of resistant bacteria, or even the distribution function itself. The moments are comparatively easy to obtain, while the calculation of the distribution function involves considerable mathematical difficulties. An

7 In some of the experiments reported in the present paper we did not

determine the total number of resistant bacteria in each culture, but the number contained in a small sample from each culture. In these cases the variance of the distribution of the number of resistant bacteria will be slightly increased by the sampling error. The proper procedure is here first to find the average number of resistant bacteria per culture by multiplying the average per sample by the ratio

(13) volume of culture

volume of sample

second, to evaluate the mutation rate with the help of equation (8); third, to figure the likely variance for the cultures by equation (11); fourth, to divide this variance by the square of the ratio (13) to obtain that part of the variance in the samples which is due to the chance distribution of the mutations. The experimental variance should be greater than this value, on account of the sampling variance. The sampling variance is in all our cases only a small correction to the total variance, and it is sufficient to use its upper limit, that of the Poisson distribution, in our calculations. Consequently, when comparing the experimental with the calculated values, we first subtract from the experimental value the sampling variance, which we take to be equal to the average number of resistant bacteria.



approximation to the beginning of the distribution function — that is, to its values for small numbers of resistant bacteria — may be obtained by grouping mutations according to the bacterial generation during which they occurred. For instance, the probability of obtaining seven resistant bacteria may be broken down into the sum of the following alternative events: (a) seven mutations during the last generation; (b) three mutations during the last generation and two mutations one generation back; (c) three mutations during the last generation and one mutation two generations back; (d) one mutation during the last generation and three mutations one generation back; (e) one mutation during the last generation, one mutation one generation back and one mutation two generations back. The probability of each of these events depends only on the mutation rate and on the final number of bacteria. The grouping of mutations according to the bacterial generation during which they occurred, and the assumption that the bacteria increase in simple geometric progression, simplify the calculation sufficiently to permit numerical computation. On the other hand, the classes with two, four, eight, etc., mutants are artificially favored by this procedure, so that a somewhat uneven distribution results, with too high values for two, four, eight, etc., resistant bacteria (see fig. 2).

MATERIAL AND METHODS

The material used for our experimental study consisted of a bacterial virus α and of its host, Escherichia coli B (DELBRÜCK and LURIA 1942). Secondary cultures after apparently complete lysis of B by virus α show up within a few hours from the time of clearing. They consist of cells which are resistant to the action of virus α, but sensitive to a series of other viruses active on B. The resistant cells breed true and can be established easily as pure cultures. No trace of virus could be found in any pure culture of the resistant bacteria studied in this paper. The resistant strains are therefore to be considered as non-lysogenic. Tests were made to see whether the resistance to virus α was a stable character of the resistant strains. In the first place, it was found that virus α is not appreciably adsorbed by any of the resistant strains. In the second place, when a certain amount of virus α is mixed with a growing culture of a resistant strain, no measurable increase of the titer of virus α occurs over a period of several hours. This is a very sensitive test for the occurrence of sensitive bacteria, and its negative result for



all resistant strains shows that reversion to sensitivity must be a very rare event. Morphologically at least two types of colonies of resistant bacteria may be distinguished. The first type of colony is similar to the type produced by the sensitive strain both in size and in the character of the surface and of the edge. The second type of colony is much smaller and translucent. The difference in colony type is maintained in subcultures. Microscopically the bacteria from these two types of colonies are indistinguishable. They also do not differ from each other or from the sensitive strain in their fermentation reactions on common sugars and in the characteristics of their growth curves in nutrient broth. In particular, the lag periods, the division times during the logarithmic phase of growth and the maximum titers attained are identical for the sensitive strain and for the two variants. Both variants, therefore, fulfill the requirements for the applicability of the theory developed above. In the presentation of our experimental results we have lumped the counts of the two types of colonies together, because: (1) theoretically, this is equivalent to summing the corresponding mutation rates; (2) experimentally, we are not certain whether each of these types does not actually comprise a diversity of variants; (3) experimentally, no correlation appeared to exist between the occurrence of these variants, which shows the independence of the causes of their occurrence. Cultures of B were grown either in nutrient broth (containing .5 percent NaCl) or in an asparagin-glucose synthetic medium. In the latter, the division time during the logarithmic phase of growth was 35 minutes, as compared with 19 minutes in broth. In synthetic medium, the acidity increased during the time of incubation from pH 7 to pH 5. In cultures of strain B, between 10-8 and 10-5 of the bacteria are found usually to give colonies resistant to the action of virus α when samples of such cultures are plated with large amounts of virus. In order to be reasonably certain that the resistant bacteria found in the test had not been introduced into the test culture with the initial inoculum, the test cultures were always started with very small inocula, containing between 50 and 500 bacteria from a growing culture. Thus any resistant bacterium found at the moment of testing (when the culture contains between 108 and 5 × 109 bacteria/cc) must be an offspring of one of the sensitive bacteria of the inoculum. All platings were made on nutrient agar plates. The plating experiments for counting the number of resistant bacteria in a liquid culture of the sensitive strain were done by plating either a portion or the entire culture with a large amount of virus α. The virus was plated first, and spread over the entire surface of the agar. A few minutes later the bacterial suspension to be tested was spread over the central part of



the plate, leaving a margin of at least one centimeter. Thus all bacteria were surrounded by large numbers of virus particles. Microscopic examination of plates seeded in this manner showed that lysis takes place very quickly; only bacteria which at the time of plating were in the process of division may sometimes complete the division. The resistant colonies which appear after incubation are therefore due to resistant bacterial cells present at the time of plating. The total number of bacteria present in the culture to be tested was deter-mined by colony counts in the usual manner. The resistant colonies of the large type appear after 12–16 hours of incubation, the colonies of the small type appear after 18–24 hours, and never reach half the size of the former ones. Counts were usually made after 24 and 48 hours.

EXPERIMENTAL

A Test of the Reliability of the Plating Method

In our experiments we wanted to study the fluctuations of the numbers of resistant bacteria found in cultures of sensitive bacteria. It was therefore necessary to show first that the method of testing did not involve any unrecognized variables, which caused the number of resistant colonies to vary from plate to plate or from sample to sample. Therefore, parallel platings were made using a series of samples from the same bacterial culture. If our plating method is reliable, fluctuations should in this arrangement be due to random sampling only, and the variance from a series of such samples should be equal to the mean. Table I gives the results of three such experiments. It will be seen that in all three cases variance and mean agree as well as may be expected. There is therefore no reason to assume that the method of sampling or plating introduces any fluctuations into our results besides the sampling error.

Fluctuations of the Number of Resistant Bacteria in Samples from a Series of Similar Cultures

As pointed out in the introduction and in the theoretical part, the hypothesis of acquired immunity and the hypothesis of mutation lead to radically different predictions regarding the distribution of the number of resistant bacteria in a series of similar cultures. The hypothesis of acquired immunity predicts a variance equal to the average, as in



sampling, while the mutation hypothesis predicts a much greater variance.

TA B L E 1

The number of resistant bacteria in different samples from the same culture.

1

Sample Number Exp. No. 10aResistant Colonies

14

Exp. No. 11aResistant Colonies

46

Exp. No. 3Resistant Colonies

42 15 56 23 13 52 24 21 48 15 15 65 56 14 44 27 26 49 48 16 51 29 20 56 4

10 13 47 7

mean 16.7 51.4 3.3variance 15 27 3.8

Χ2 9 5.3 12P .4 .8 .2

Series of five to 100 cultures were set up in parallel with small equal inocula, and were grown until maximum titer was reached. Three kinds of cultures were used — namely: (1) 10.0 cc aerated broth cultures; (2) .2 cc broth cultures; (3) .2 cc synthetic medium cultures. The results of all tests for the number of resistant bacteria are summarized in table 2 and table 3. It will be seen that in every experiment the fluctuation of the numbers of resistant bacteria is tremendously higher than could be accounted for by the sampling errors, in striking contrast to the results of plating from the same culture (see table 1) and in conflict with the expectations from the hypothesis of acquired immunity. We want to see next whether these results fit the expectations from the hypothesis of mutation. We must therefore compare the experimental results with the relations developed in the theoretical part, keeping in mind that the theory contains several simplifying assumptions.



TA B L E 2

The number of resistant bacteria in series of similar cultures.

1

Experiment No.

23456789

1011121314151617181920

Culture No.

1

1018

125101427

317

Number of CulturesVolume of Cultures, ccVolume of Samples, cc

9 10.0

.05

10

294117203130

717

8 10.0

.05

11

30

10 10.0

.05

15

6

10 10.0

.05

16

1

20 .2* .08

17

1

12

.08

21a

0

19

.05

21b

38

5 10.0

.05.2* .2

10 5 0 0 0 2840 10 3 0 0 3545 8 0 7 0 107

183 24 0 0 8 1312 13 5 303 1

173 165 0 0 023 15 5 0 157 6 0 3 051 10 6 48 15

107 1 00 4 00 190 01 00 170 11

64 00 0

33

26.8 23.8 62 26.2 11.35 48.230 3.8Average per sample

1217 84 3498 2178 694 11726620 40.8Variance (corrected for sampling)

5360 4760 12400 5240 28.4 844075 15.1Average per culture

Bacteria per culture 3.4×1010 4×1010 4×1010 2.9×1010 5.6×108 5×108 1.1×108 3.2×1010

Mutation rate 1.8×10-8 1.4×10-8 4.1×10-8 2.1×10-8 1.1×10-8 3.0×10-8 3.3×10-8 3.0×10-8

1.3 .39 .95 1.8 2.3 .712.7 1.7Standard deviation

.35 .33 .33 .37 .94 .26.67 1.04 Average

exp.

calc

* Cultures in synthetic medium

First we can compare, according to equation (12), the experimental and the calculated values of the ratio between the standard deviation and the average of the numbers of resistant bacteria. These ratios are included in tables 2 and 3. It is seen that the experimental and theoretical values are reasonably close. However, in all but one case the experimental ratio is greater than the value calculated from the theory — that is, the variability is even greater than predicted.



TA B L E 3

Distribution of the numbers of resistant bacteria in series of similar cultures.

Experiment No.

Resistantbacteria

Number of CulturesVolume of Cultures, ccVolume of Samples, cc

22

100

.05

57

23

87

.2.2*

2052317220

10.12Average per sample

6270Variance (corrected for sampling)

40.48Average per culture

Bacteria per culture 2.8×108

Mutation rate 2.3×10-8

7.8Standard deviation

1.5 Average

exp.

calc


.2*

Number ofcultures

001

0123456

112151

101201501

102050

100200500

1000

-------

Resistantbacteria

2917

43325675

Number ofcultures

241

0123456

112151

101201501

102050

100200500

1000

-------

28.6

6431

28.6

2.4×108

2.37×10-8

2.8

1.5

A part of this discrepancy may be accounted for by the fact that the time t0, mutations occurring prior to which were disregarded by the theory, was chosen in such a manner that on the average one mutation would occur prior to time t0. This mutation, if it occurs, will of course tend to increase the variance, and in some of the experiments the high value of the experimental variance can be traced directly to one exceptional culture in which a mutation had evidently occurred several generations prior to time t0. Unfortunately, there is no general criterion by which one might eliminate such cultures from the statistical analysis, because, in a culture with an exceptionally high count of resistant bacteria, these do not necessarily stem from one exceptionally



early mutation, but may also be due to an exceptionally large number of mutations after time t0. There may also be other reasons why the observed variances are higher than the expected ones. First of all, the simplifying assumption that the mutation rate per bacterial generation is independent of the physiological state of the bacteria may be too simple. If the mutation rate is higher for actively growing bacteria than for bacteria near the saturation limit of the cultures, early mutations and big clone sizes will be favored, and therefore higher variations of the numbers of resistant bacteria can be expected. Second, the assumption of a sudden transition from sensitivity to resistance may also be too simple. It is conceivable that the character “resistance to virus” may not fully develop in the bacterial cell in which the mutation occurs, but only in its offspring, after one or more generations. However, if this were the case, cultures with only one or two resistant bacteria should be relatively rare. The last experiment listed in table 3, in which the entire cultures were plated, shows a rather high proportion of cultures with only one resistant bacterium. This seems to show that the character “resistance to virus” in general does come to expression in the bacterial cell in which the corresponding mutation occurred, as assumed by the theory.

10

20

30

40

50

1 2 3 4 5 6 7 8 9 >9 ResistantBacteria

Cultures%

0

Figure 2. — Experimental (Experiment No. 23) and calculated distributions of the numbers of resistant bacteria in a series of similar cultures. Solid columns: experimental. Stippled columns: calculated.

Another way of comparing the experimental results with the theory is to compare the experimental distribution of resistant bacteria with the approximate distribution calculated by the method outlined at the end of the theoretical part. The theoretical distribution has to be calculated from the average number of mutations per culture given by equation (5). Only experiments where the whole culture is tested can therefore be used for such a comparison. This method tests the fitting of the



expectations for small numbers of resistant bacteria, in contrast to the comparison of the standard deviations, which involves predominantly the cultures with high numbers of resistant bacteria. Figure 2 shows the experimental and calculated distributions for Experiment No. 23; the cultures with more than nine resistant bacteria are lumped together in one class, since the distribution has not been calculated for values higher than nine. It is seen that the fitting for small values is satisfactory. In particular, the number of cultures with one resistant bacterium very closely fits the expectation. The classes with two, four, eight, etc., resistant bacteria are bound to be favored in the theoretical distribution, as explained in the theoretical part. The results shown in figure 2 also confirm the assumption that the discrepancy between experimental and calculated standard deviations must be due to an excess of cultures with large numbers of resistant bacteria. Summing up the evidence, we may say that the experiments show clearly that the resistant bacteria appear in similar cultures not as random samples but in groups of varying sizes, indicating a correlating cause for such grouping, and that the assumption of genetic relatedness of the bacteria of such groups offers the simplest explanation for them.

Mutation Rate

As pointed out in the theoretical part of this paper, mutation rates may be estimated from the experiments by two essentially different methods. The first method makes use of the fact that the number of mutations in a series of similar cultures should be distributed in accordance with Poisson’s law; the average number of mutations per culture is calculated from the proportion of cultures containing no resistant bacteria at the moment of the test, according to equation (5). There are two technical difficulties involved in the application of this method. In the first place, rather large numbers of cultures have to be handled and conditions have to be chosen so that the proportion of resistant bacteria is neither too small nor too large. In the second place, the entire cultures have to be tested, which means, in our method of testing, that cultures of rather small volume have to be used and great care must be taken to plate as nearly as possible the entire culture. Experiment No. 23 (see table 3) permits an estimate of the mutation rate by this method. Out of 87 cultures, no resistant bacteria were found in 29 cultures, a proportion of .33. From equation (5) we calculate therefore that the average number of mutations per culture in this experiment was 1.10. Since the total number of bacteria per culture was 2.4 × 108, we obtain as the mutation rate, from equation (4),



a = .47 × 10-8 mutations per bacterium per time unit = .32 × 10-8 mutations per bacterium per division cycle. This calculation makes use exclusively of the proportion of cultures containing no resistant bacteria. It is therefore inefficient in its use of the information gathered in the experiment. The second method makes use of the average number of resistant bacteria per culture. The relation of this average number with the mutation rate was discussed in the theoretical part of this paper and was found to be expressed by equation (8). The mutation rates calculated by this method for each experiment are collected in table 4.

TA B L E 4

Values of mutation rate from different experiments.

1

Experiment No. Number of Cultures

9

Volume of Cultures

10.0

Mutation Rate

10 8 10.011 10 10.015 10 10.016 2017 12

21a 19 .221b 5 10.0

22 10023 87 .2*

average 2.45 ×10-8

.2*

.2*

.2*

2.4 ×10-8

2.3 ×10-8

3.0 ×10-8

3.3 ×10-8

3.0 ×10-8

1.1 ×10-8

2.1 ×10-8

4.1 ×10-8

1.4 ×10-8

1.8 ×10-8

cc Mutation per bacteriumper time unit


It will he seen that the values of the mutation rate obtained by the second method are all higher than the value found by the first method. This discrepancy may be traced back to the same cause as the discrepancy between the calculated and observed values of the standard deviation of the numbers of resistant bacteria. This, we found, was due to an excess of early mutations, giving rise to big clones of resistant bacteria. These big clones do not affect the mutation rate calculated by



the first method, but they do affect the results of the second method, which is based on the average number of resistant bacteria. One sees in table 4 that the mutation rate calculated by the second method does not vary greatly from experiment to experiment. In particular, it will be noted that there is no significant difference between the values obtained from cultures in broth and from cultures in synthetic medium, notwithstanding the considerable difference of metabolic activity and of growth rate of the bacteria in these two media. This shows that the simple assumption of a fixed small chance of mutation per physiological time unit is vindicated by the results. It may also be noted in table 4 that there is no significant difference between the mutation rates obtained from 10 cc cultures and those obtained from .2 cc cultures, or between the experiments with many and those with few cultures. The variability of the value of the mutation rate seems to be solely due to the peculiar probability distribution of the number of resistant bacteria in series of similar cultures predicted by the mutation theory. At this point an experiment may be mentioned by which it was desired to find out whether or not mutations occur in a culture after the bacteria have ceased growing. A culture was grown to saturation and was then tested repeatedly for resistant bacteria and for total number of bacteria over several days. The proportion of resistant bacteria did not change, even when the sensitive bacteria began to die, showing that the resistant bacteria have the same death rate in aging cultures as the sensitive bacteria.

DISCUSSION

We consider the above results as proof that in our case the resistance to virus is due to a heritable change of the bacterial cell which occurs independently of the action of the virus. It remains to be seen whether or not this is the general rule. There is reason to suspect that the mechanism is more complex in cases where the resistant culture develops only several days after lysis of the sensitive bacteria. The proportion of mutant organisms in a culture and the mutation rate are far smaller in our case than in other studied cases of heritable bacterial variation. The possibility of investigation of such rare mutations is in our case merely the result of the method of detecting the mutant organisms. In other cases, the variants are detected by changes in the colony type which is produced by the mutant organism, either in the pigmentation or in the character of the surface or the edge of the colony. Often, colonies of intermediate character occur, and it is



difficult to decide whether they are mixed colonies or stem from bacteria with intermediate character. This is particularly true of cases where the mutation rate is high and where reverse mutation occurs. Fairly high mutation rates, however, are a prerequisite of any study of colony variants, since the number of colonies that can be examined is limited by practical reasons. The study of mutations causing virus resistance is free of these difficulties. The segregation of the mutant from the normal organisms occurs in the one-cell stage by elimination of the normal individuals, and the character of the colony which develops from a mutant organism is of secondary importance. Owing to the total elimination of the normal individuals, the number of organisms which may be examined is very much higher than for any other method; more than 108 bacteria may be tested on a single plate. Since the mutations to virus resistance are often associated with other significant characters, the method may well assume importance with regard to the general problems of bacterial variation. It must not be supposed that the peculiar statistical difficulties encountered in our case are restricted to cases of very low mutation rates. The essential condition for the occurrence of the peculiar distribution studied in the theoretical part of this paper is the following: the initial number of bacteria in a culture must be so small that the number of mutations which occur during the first division cycle of the bacteria is a small number. This will always be true, however great the mutation rate, if one studies cultures containing initially a small number of organisms. In a series of very interesting studies of the color variants of Serratia marcescens, Bunting (1940a, 1940b, 1942; Bunting and Ingraham 1942) succeeded to some extent in obviating the statistical difficulties by always using inocula of about 100,000 bacteria. In some of her cases this number was sufficiently high to result in numerous mutations during the first division cycle of the bacteria. In other cases the number was apparently not high enough, since the author reports troublesome variations of the fractions of variants in successive subcultures. In those cases where the size of the inocula was high enough, the author succeeded in deriving reproducible values for the mutation rates from the study of single cultures, followed through numerous subcultures. In these cases it is sufficient to apply the equations of the theory referring to the average numbers of mutants as a function of time. It is clear, however, that this method is applicable only in cases of mutation rates of at least 10-4 per bacterium per division cycle.



In our case, as in many others, the virus resistant variants do not exhibit any striking correlated physiological changes. There is therefore little opportunity for an inquiry into the nature of the physiological changes responsible for the resistance to virus. Since the offspring of the mutant bacteria, when isolated after the test, are unable to synthesize the surface elements to which the virus is specifically adsorbed in the sensitive strain, one might suppose that this loss is a direct effect of the mutation. However, it is also conceivable that the loss occurs upon contact with virus, since it is detected only after such contact (hypothesis bl). In some of the cases studied by BURNET (1929), where the mutational change to resistance is correlated with a change of phase, from smooth to rough or vice versa, the change of the surface structure must be a direct result of the mutation, since the mutant colonies may be picked up prior to the resistance test and, when tested, exhibit the typical change of affinity of the surface structure. These findings make it more probable that the loss of surface affinity to virus is a direct effect of the mutation. The alteration of specific surface structures due to genetic change is a phenomenon of the widest occurrence. The genetic factors determining the antigenic properties of erythrocytes are well known. There is evidence (WEBSTER 1937; HOLMES 1938; STEVENSON, SCHULTZ, and CLARK 1939) that resistance or sensitivity to virus in plants and animals is correlated with, or even dependent on, genetic changes, possibly affecting the antigenic make-up of the cellular surface. The proof that resistance to a bacterial virus may be traced to a specific genetic change may assume importance, therefore, with regard to the general problems of virus sensitivity and virus resistance.

SUMMARY

The distribution of the numbers of virus resistant bacteria in series of similar cultures of a virus-sensitive strain has been analyzed theoretically on the basis of two current hypotheses concerning the origin of the resistant bacteria. The distribution has been studied experimentally and has been found to conform with the conclusions drawn from the hypothesis that the resistant bacteria arise by mutations of sensitive cells independently of the action of virus. The mutation rate has been determined experimentally.



LITERATURE CITED

BUNTING, M. I., 1940a A description of some color variants produced by Serratia marcescens, strain 274. J. Bact. 40: 57–68.

1940b The production of stable populations of color variants of Serratia marcescens #274 in rapidly growing cultures. J. Bact. 40: 69–81.

1942 Factors affecting the distribution of color variants in aging broth cultures of Serratia marcescens #274. J. Bact. 43: 593–606.

BUNTING, M. I., and L. J. INGRAHAM, 1942 The distribution of color variants in aging broth cultures of Serratia marcescens #274. J. Bact. 43: 585–591.

BURNET, F. M., 1929 Smooth-rough variation in bacteria in its relation to bacteriophage. J. Path. Bact. 32: 15–42.

DELBRÜCK, M., and S. E. LURIA, 1942 Interference between bacterial viruses. I. Arch. Biochem. 1: 111–141.

GRATIA, A., 1921 Studies on the d’Herelle phenomenon. J. Exp. Med. 34: 115–131.

D’HERELLE, F., 1926 The Bacteriophage and Its Behavior. Baltimore: Williams and Wilkins.

HOLMES, F. O., 1938 Inheritance of resistance to tobacco-mosaic disease in tobacco. Phytopathology 28: 553–561.

STEVENSON, F. J., E. S. SCHULTZ, and C. F. CLARK, 1939 Inheritance of immunity from virus X (latent mosaic) in the potato. Phytopathology 29: 362–365.

WEBSTER, L. T., 1937 Inheritance of resistance of mice to enteric bacterial and neurotropic virus infections. J. Exp. Med. 65: 261–286.