Genetics: Published Articles Ahead of Print, published on July 14, 2005 as 10.1534/genetics.104.035774
Update on Estimation of Mutation Rates Using Data from Fluctuation Experiments Qi Zheng Department of Epidemiology and Biostatistics, School of Rural Public Health Texas A & M University System Health Science Center, Bryan, Texas 77802 Phone: (979)845-4379 E-mail:
[email protected] ABSTRACT This note discusses a minor mathematical error and a problematic mathematical ¨ck’s (1943) classic paper on fluctuation analassumption in Luria and Delbru ysis. In addition to suggesting remedial measures, the note provides information on the latest developments of techniques for estimating mutation rates using data from fluctuation experiments. ¨ck (1943) still serves as The fluctuation test protocol devised by Luria and Delbru the basis for estimating microbial mutation rates today, although later developments have resulted in much improved methods. Among the later contributions that enhanced our ability to analyze fluctuation experiments are those made by Lea and Coulson (1949), Armitage (1952), Crump and Hoel (1974), Mandelbrot (1974), Koch (1982), Stewart et al. (1990), Ma et al. (1992), Jones et al. (1994) and many others found in a recent review (Zheng, 1999). Rosche and Foster’s (2000) critical comparison of the then existing methods is a useful guide for biologists. One goal of this note is to notify the reader of the latest developments that can further help biologists improve their ability of measuring mutation rates. Another goal is to discuss a minor mathematical error and a problematic ¨ck’s (1943) paper that have caused lingermathematical assumption in Luria and Delbru ing confusion. Previous attempts to clarify the confusion were scarce and to a large extent failed to resolve some relevant practical issues. As a result, the genetics literature is increasingly fraught with mutation rates that were computed using either incorrect or unreliable methods. It appears helpful that the minor error and the problematic assumption be ex1
plained and remedial measures be provided. I shall begin with a paradox that has puzzled many. In a fluctuation experiment, each of n parallel cultures is seeded at time zero with N0 nonmutant cells for incubation. At a later time T each culture has about NT nonmutant cells and the contents of each culture are plated to facilitate counting of mutants existing at time T in the n cultures. This process results in experimental data in the form of X1 , X2 , . . . , Xn , the numbers of mutants existing in the n cultures immediately before plating. If z of the n ¨ck’s (1943) P0 cultures still remain devoid of mutant cells at time T , Luria and Delbru method estimates the mutation rate by µ ˆ0 = log 2
− log(ˆ p0 ) N T − N0
(1)
¨ck did not give the above equation, their with pˆ0 = z/n. Although Luria and Delbru numerical example on page 507 clearly indicates that they used (1) to estimate mutation rates defined as “mutations per bacterium per division cycle.” This definition of mutation rates has been widely accepted, and throughout this note the term “mutation rate” is used in that sense. In other words, a mutation rate is the probability that a cell undergoes a mutation during the cell’s life cycle. Using the same definition, Lederberg (1951, p. 99) argued that mutation rates should be estimated by µ ˆ0 =
− log(ˆ p0 ) . NT − N0
(2)
Lederberg’s reasoning runs as follows. In each culture NT − N0 cellular divisions have happened. If µ0 is the mutation rate, the probability that a culture is devoid of mutants after NT − N0 cellular divisions is P0 = (1 − µ0 )NT −N0 ≈ exp(−µ0 (NT − N0 )). Equating P0 with pˆ0 yields an estimator of µ0 in the form of (2). Thus equations (1) and (2), differing by a factor of log 2, aim at estimating the same quantity. 2
This paradox has led some to seek justifications of (1) (e.g., Hayes 1968, p. 194 and Drake 1970, p. 49), and others to cast doubt on it (e.g., Lea and Coulson 1949, p. 266 and Kondo 1972). No consensus has emerged. A helpful approach to clarifying this issue is to present an argument that not only reinforces the validity of (2) but also highlights what was ¨ck in arriving at (1). Note that Luria and Delbru ¨ck overlooked by Luria and Delbru used average doubling time divided by log 2 to measure time, rendering their derivation of (1) unnecessarily difficult to understand. I shall use clock time by introducing an explicit cell growth parameter β. Specifically, the nonmutant population size at time t, denoted by N (t), will be modeled by an exponential function N (t) = N0 eβt . Thus, occurrence of mutation is assumed to be governed by a Poisson process having intensity function µN0 eβt . Because the probability of a mutation occurring in a small time interval (t, t + ∆t) is approximately µN0 eβt ∆t, the parameter µ is often called the probability of mutation per cell per unit time. However, except for time points t = 0 and t = tk with tk = β −1 log(k/N0 ) for k = 1, 2 . . ., N (t) = N0 eβt does not represent the actual population size at time t, because N (t) is a positive integer if and only if t = 0 or t = tk for some k. A literal interpretation of µ as “mutation per cell per unit time” out of the intended context can lead to unexpected results, as the following analysis will demonstrate. Consider a time interval (tk , tk+1 ] for an arbitrary positive integer k. As hinted above, tk+1 − tk can be viewed as an inter-division time under the assumption N (t) = N0 eβt . In view of the Poisson mutation model, the probability of one or more mutations occurring in that time interval is Z 1 − exp −
tk+1
tk
µ µ ≈ . µN0 e dt = 1 − exp − β β βt
(3)
Notably, this probability is independent of k, which suggests viewing µβ = µ/β as the probability of one or more mutations occurring between two consecutive cellular divisions.
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Since µβ is presumably small, the probability of two or more mutations occurring in (tk , tk+1 ] is negligible compared to the probability of exactly one mutation happening in that interval. Therefore, µβ can be regarded as the probability of a mutation per cell division, or the mutation rate. Note that the expected number of mutations accumulated by time T is Z T m(T ) = µN0 eβs ds = µβ (NT − N0 ). (4) 0
Let p0 (T ) be the probability that no mutation occurs in a culture by time T . In light of the Poisson mutation model, one has p0 (T ) = e−m(T ) . Therefore, it follows from (4) that µβ =
− log(p0 (T )) m(T ) = . NT − N0 NT − N0
(5)
Replacing p0 (T ) with pˆ0 leads to an estimator identical to that given in (2). On the other ¨ck’s reasoning leading to (1) was based on their interpretation of hand, Luria and Delbru µ as “a fixed small chance per unit time for each bacterium to undergo a mutation.” (Luria ¨ck, 1943, p. 494) Because a cell’s life cycle is considered to be (log 2)/β under and Delbru the assumption N (t) = N0 eβt , µ×
log 2 = (log 2)µβ β
(6)
seems to be the probability of mutation per cell division, namely, the mutation rate. However, (6) is merely a literal interpretation of µ as the probability of mutation per cell per unit time; the precise meaning of µ can be grasped only within the context of the important assumption that occurrence of mutation is governed by an inhomogeneous Poisson process having an “instantaneous” rate µN0 eβt at time t. Because (6) deviates from the intended definition of µ, the aforementioned paradox results. The above discussion indicates that, from a theoretical point of view, one should choose µβ as a definition of the mutation rate, and use (2) instead of (1) as an estimator thereof. However, the discussion does not answer the question of whether µβ agrees with the definition of a mutation rate from a practical point of view. To address this issue I ran several 4
simulations, one of which I now report. I first simulated the numbers of mutants for 30,000 cultures. To maintain a degree of independence between my simulation and the mathematical model that gives rise to (5), I used a discrete-time cell division model based on the five assumptions given by Angerer (2001, p. 149). Specifically, each culture starts with one nonmutant cell. At each step one cell from the culture is chosen to divide. If the culture already has x mutants and y nonmutants, then the probability that a mutant is chosen to divide is x/(x + y), and the probability that a nonmutant is chosen to divide is y/(x + y). When a mutant divides, it splits into two mutant daughter cells; when a nonmutant divides, it splits into one mutant and one nonmutant with probability p, or it splits into two nonmutants with probability 1 − p. (Clearly, p thus defined agrees with the definition of a mutation rate as is commonly understood.) For each culture, this procedure is repeated until x + y = NT . In the simulation I chose p = 5 × 10−8 and NT = 108 . I then considered the first 30 cultures as coming from the first experiment, the next 30 cultures as coming from the second experiment, and so on. For each simulated experiment, I used SALVADOR (Zheng 2002 and 2005) to find the maximum likelihood estimate of m(T ). Finally, I used the relation µβ ≈ m(T )/NT suggested by (5) to compute mutation rates. The average of these 1,000 estimated mutation rates is 5.036 × 10−8 and the median is 4.988 × 10−8 , indicating that µβ , and hence not (log 2)µβ , coincides with p. The distribution of these 1,000 estimates of the mutation rate is summarized in Figure 1. Another issue that has also caused lingering confusion is the use of the sample mean ¯ has too large a ¯ = n−1 Pn Xi in estimating mutation rates. Because the sample mean X X i=1 ¨ck introduced the concept variance to be useful in statistical inference, Luria and Delbru of a “likely average” to alleviate this problem. Theoretically, the expected number of mutants ¨ck’s (1943) equation (6), in a culture can be found by solving Luria and Delbru dρ(t) = βρ(t) + µN0 eβt , dt 5
(7)
with the initial condition ρ(0) = 0. From standard differential equation theory it follows that βt
Z
ρ(t) = µN0 e
t
ds = µtN0 eβt .
(8)
0
Thus, the expected number of mutants in a culture at time T is ρ = ρ(T ) = µT N0 eβT .
(9)
¨ck reasoned that prior to certain time t0 mutation is unlikely to occur Luria and Delbru in an experiment. Therefore, they changed the lower limit of integration in (8) from zero to some t0 > 0, giving their equation (6a), ρ˜(t) = (t − t0 )µN0 eβt .
(10)
¨ck chose t0 to be the epoch at which one would expect the first Because Luria and Delbru mutation to occur among the n cultures, they replaced T in (4) with t0 to yield m(t0 ) = µβ N0 (eβt0 − 1) =
1 . n
(11)
¨ck further assumed that eβt0 − 1 ≈ eβt0 ; as a result, (11) reduces to Luria and Delbru nµβ N (t0 ) = 1.
(12)
On the other hand, rearranging NT = N (t0 )eβ(T −t0 ) gives β(T − t0 ) = log(NT /N (t0 )). Because N (t0 ) = 1/(nµβ ) from (12), it then follows that T − t0 =
1 log(nµβ NT ), β
(13)
¨ck’s equation (7). Combining (13) and (10) yields an expression which is Luria and Delbru ¨ck’s equation (8): for the likely average equivalent to Luria and Delbru ρ˜ = ρ˜(T ) = µβ NT log(nµβ NT ). 6
(14)
By equating the above likely average with a sample mean one can estimate µβ by numerically solving ¯ = µβ NT log(nµβ NT ). X
(15)
Lederberg (1951, p. 99) observed that this method “offers certain short-term advantages,” ¯ Recently but did not satisfactorily solve the intrinsic problem due to the high variability of X. Lederberg’s caution began to be appreciated, and (15) is no longer in common use. But the idea of a likely average has become entrenched in the literature due to a popular ¯ with the sample modification of (15). Setting n = 1 and replacing the sample mean X median ξˆ0.5 yields an estimating equation ξˆ0.5 = µβ NT log(µβ NT ),
(16)
which is often called Drake’s formula. Thus, Drake’s formula is based on the concept of a modified likely average — the quantity t0 is increased to the epoch at which one would expect the first mutation to occur in a single culture. As Rosche and Foster’s (2000) simulations indicate, Drake’s formula is good only when the expected number of mutations per culture, m(T ), is about 30, a rare experimental scenario. The inadequate performance of (16) casts doubt on the usefulness of the concept of a likely average. In assessing the usefulness of a likely average, one might note that (10) is a solution of (7) satisfying the initial condition ρ(t0 ) = 0. Therefore, the likely average ρ˜ in (14) can be regarded as the mean number of mutants in a culture under the assumption that cells were prevented from mutating before an arbitrarily chosen time t0 . It seems a problematic assumption that a likely average should be superior to the exact average ρ given in (9). The concept of a “likely average” has neither theoretical nor empirical bases, and hence its usefulness in estimating mutation rates is dubious at best. Now I suggest the following guidelines for mutation rate estimation. First, in the context 7
of fluctuation experiments, the term mutation rate should be reserved for µβ , because this quantity agrees with the accepted definition of a mutation rate as the probability that a cell undergoes a mutation during its life cycle. The parameter µ is a necessary mathematical devise, but the term “mutation rate per cell per unit time” can be avoided in most biological contexts to avoid confusion. Second, published mutation rates computed using (1) should be divided by log 2; the P0 method is still needed when the number of mutants in a culture is difficult to ascertain, but the presence or absence of mutants in a culture can be determined. In applying the P0 method one should use (2). Third, the concept of a “likely average” is obsolete, and so are methods based on that concept, e.g., equations (15) and (16). If a published mutation rate was computed using (16), one can use the same equation to recover the sample median ξˆ0.5 (provided NT is known); a more reliable estimate of the mutation rate can then be obtained by using Lea and Coulson’s (1949) method of the median or by applying equation (6) of Jones et al. (1994), m(T ˆ )=
ξˆ0.5 − log 2 , log(ξˆ0.5 ) − log(log 2)
to estimate m(T ) from ξˆ0.5 . The mutation rate µβ can then be extracted from the relation µβ = m(T )/(NT − N0 ) as given in (5). Finally, as Rosche and Foster (2000) emphasized, if all n observations X1 , . . . , Xn from a fluctuation experiment are available, the best approach for estimating a mutation rate is to use the maximum likelihood method to estimate m(T ). Use of the maximum likelihood method was not common in the past, partly due to lack of convenient and efficient computer software written specifically for fluctuation analysis. This situation has been ameliorated by the appearance of SALVADOR, which includes most of the existing methods for fluctuation analysis. In particular, SALVADOR provides methods for analyzing experiments where mutants and nonmutants grow at different rates. Moreover, recent theoretical developments (Zheng 2002 and 2005) have
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made it possible to construct asymptotic confidence intervals for mutation rates. These interval estimation methods can be readily applied via SALVADOR, which is available at http://library.wolfram.com/infocenter/MathSource/5556.
I am much indebted to two anonymous reviewers whose detailed comments substantially improved the presentation.
9
125
Frequency
100
75
50
25
4×10−8
5×10−8
6×10−8
7×10−8
Estimated mutation rate
Figure 1 Distribution of 1,000 estimated mutation rates obtained by simulating fluctuation experiments. Each simulated experiments consists of 30 cultures, and the probability of mutation per cell division is 5 × 10−8 . The estimated µβ are found to fluctuate around the true mutation rate, indicating that (log 2)µβ does not represent the mutation rate as is commonly understood.
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LITERATURE CITED 1. Angerer, W. P., 2001 An explicit representation of the Luria-Delbr¨ uck distribution. J. Math. Biol. 42: 145-174. 2. Armitage, P., 1952 The statistical theory of bacterial populations subject to mutation, J. R. Statist. Soc. B 14: 1-33. 3. Crump, K.S. and D.G. Hoel, 1974 Mathematical models for estimating mutation rates in cell populations. Biometrika 61: 237-252. 4. Drake, J. W., 1970 The Molecular Basis of Mutation, Holden-Day, San Francisco. 5. Hayes, W. 1968 The Genetics of Bacteria and their Viruses: Studies in Basic Genetics and Molecular Biology, 2nd Edition, Wiley, New York. 6. Jones, M. E., S. M. Thomas and A. Rogers, 1994 Luria-Delbr¨ uck fluctuation experiments: design and analysis. Genetics 136: 1209-1216. 7. Koch, A.L., 1982 Mutation and growth rates from Luria-Delbr¨ uck fluctuation tests. Mutat. Res. 95: 129-143. 8. Kondo, S., 1972 A theoretical study on spontaneous mutation rate. Mutat. Res. 14: 365-374. 9. Lea, E. A. and C. A. Coulson, 1949 The distribution of the numbers of mutants in bacterial populations. J. Genetics 49: 264-285. 10. Lederberg, J., 1951 Inheritance, variation, and adaptation, pp. 67-100 in Bacterial Physiology, edited by C.H. Werkman and P.W. Wilson, Academic Press, New York. 11. Luria, S. E. and M. Delbr¨ uck, 1943 Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28: 491-511. 11
12. W.T. Ma, G. vH. Sandri and S. Sarkar, 1992 Analysis of the Luria and Delbr¨ uck distribution using discrete convolution powers. J. Appl. Prob. 29: 255-267. 13. B. Mandelbrot, 1974 A population birth-and-mutation process, I: Explicit distributions for the number of mutants in an old culture of bacteria. J. Appl. Prob. 11: 437-444. 14. Rosche, W. A. and P. L. Foster, 2000 Determining mutation rates in bacterial populations. Methods, 20: 4-17. 15. Stewart, F.M., D.M. Gordon and B.R. Levin, 1990 Fluctuation analysis: The probability distribution of the number of mutants under different conditions. Genetics 124: 175-185. 16. Zheng, Q., 1999 Progress of a half century in the study of the Luria-Delbr¨ uck distribution. Mathe. Biosci. 162: 1-32. 17. Zheng, Q., 2002 Statistical and algorithmic methods for fluctuation analysis with SALVADOR as an implementation. Math. Biosci. 176: 237-252. 18. Zheng, Q., 2005 New algorithms for Luria-Delbr¨ uck fluctuation analysis. Math. Biosci., in press.
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