TY - JOUR
AU - von Haeseler, Arndt
AB - Abstract A versatile algorithm is developed to model PCR on a computer. The method is based on a modification of the coalescent process and provides a general framework to analyse data from PCR. It allows for incorporation of the dynamics of the replication process as described in terms of the number of starting template molecules and cycle-dependent PCR efficiency. The simulation method generates, as a first step, the genealogy of a set of sequences sampled from a final PCR product. In a second step a mutation process is superimposed and the resulting data set is analysed. The efficiency of our algorithm enables us to get reliable approximations of various sample distributions. We demonstrate the relevance of our method with two applications: maximum likelihood estimation of the error rate in PCR and a test of homogeneity of the template. Introduction The development of the polymerase chain reaction (PCR) has revolutionised work with genetic material (1). The ability of PCR to produce large amounts of DNA from a small number of template molecules made sequencing of DNA much faster and easier. Moreover, new applications of PCR arose in cancer research, inherited disease diagnosis, forensic medicine and ancient DNA (2). The biochemical nature of PCR led to questions about the fidelity of the copy mechanism. Experimental observations showed that replication of DNA during PCR is imperfect. Occasionally, nucleotides are substituted, added or deleted while synthesising a new DNA strand. Consequently, methods were developed to measure the fidelity of PCR. Krawczak et al. (3) and Hayashi (4) used the proportion of correctly amplified molecules in the PCR product as a measure of the error rate. Maruyama (5) gave a formula for the total number of mutations in PCR when the amount of target molecules is doubled in each cycle. This theoretically possible doubling of the target per cycle of a PCR is not observed in practice (6). Only a proportion of the template molecules is amplified in a PCR cycle. This proportion is called the efficiency λ. Using the theory of branching processes, Sun (7) and Weiss and von Haeseler (8) developed methods to estimate the error rate for constant efficiency 0 < λ ≤ 1. Their estimates are based on sequence data: after PCR the final product is cloned, a set of colonies is picked and these colonies are sequenced separately. For large numbers of initial template molecules, Sun (7) suggested a method of moments to estimate the error rate based on either the number of variable positions or the mean pairwise difference observed in a set of sequences picked from a PCR. Weiss and von Haeseler (8) gave an approximation to the pairwise difference distribution of a sample of sequences and used a least squares criterion to estimate the error rate from the observed pairwise difference distribution. They showed the robustness of the approximation even if efficiency varies from cycle to cycle, if the number of initial template molecules is moderately large. Recently, Cline et al. (9) investigated the fidelity of different DNA polymerases under various experimental conditions. Coalescent theory (10–13) is frequently used to analyse intra-population sequence data in population genetics. It describes the genealogy of a sample of individuals (or more specifically their DNA sequences) in terms of stochastic processes. This formulation facilitates the application of methods of inductive statistics to draw conclusions from a set of data (14,15). We introduce a coalescent approach to analyse data sets resulting from a PCR. After a short description of the underlying mathematical model we explain the simulations that generate sets of DNA sequences according to our model. The basic idea is as follows: in a first step the algorithm generates a genealogy that describes the relations between DNA sequences in a sample from a PCR product. According to this genealogy, DNA sequences ‘evolve’ under a specified model of the mutation process in PCR. This approach has several advantages: it is not limited to the assumption of an overall constant efficiency throughout a PCR and the number of initial template molecules is incorporated as a parameter. Thus, we are not restricted to situations where this number is very large (7). Furthermore, the efficiency of our algorithm allows us to compute reliable approximations of various sample distributions. We apply this technique to get a maximum likelihood estimation of the error rate in a PCR and to devise a statistical test that discriminates whether the initial template molecules were homogeneous or not (due to damage, heteroplasmy or contamination). We illustrate both applications with the analyses of a published data set. A Mathematical Model A model of PCR has to include a model of the replication process and the mutation process. We use the single-stranded model introduced by Krawczak et al. (3). In the following we frequently use molecule as a synonym for single-stranded sequence containing the target sequence. Any other chemical molecules that are, in reality, present in a PCR tube are not considered. The replication process of PCR is described in terms of the theory of branching processes (7,8). We assume that a PCR starts with S0 identical copies of single-stranded sequences. Let Si be the number of sequences present after the i-th cycle. In cycle i each of the Si−1 template molecules is amplified independently of the others with probability λi. The probability λi can also be viewed as the proportion of amplified molecules in cycle i, hence, it is called the efficiency in cycle i. More precisely, the efficiency does not simply depend on the cycle number, but on the number of amplifications in the previous cycles and on PCR conditions (6). If we assume that the random variable Si depends only on λi and Si−1 then the sequence S0, S1, …., Si, … forms a non-homogeneous Markov branching process (16). If λi = λ independently of the cycle number, then the accumulation of PCR product is a Galton-Watson process (7,8). Table 1 View largeDownload slide Notation summary Table 1 View largeDownload slide Notation summary Table 2 View largeDownload slide Realisation of a simulation for a single initial molecule Table 2 View largeDownload slide Realisation of a simulation for a single initial molecule Because replication in PCR is not error free, we add a mutation process to the model: we assume that a new mutation occurs at a position that has not mutated in any other sequence before. Furthermore, we model the process of nucleotide substitution as a Poisson process with parameter μ, where μ is the error rate (mutation rate) of PCR per target sequence and per replication. This so-called infinitely many sites model (17) does not allow for parallel or back mutations. In the case of PCR this assumption is reasonable because only a small number of mutations are observed in practice. However, other substitution models can be incorporated into the simulation method introduced in the following section. The Coalescent in PCR Computer simulations of stochastic processes have become a powerful tool to analyse data in situations where analytical methods are not feasible. In the population genetics literature a prominent example is the coalescent process (10,11) that describes the ancestral relationship between a sample of individuals in a population as one goes back in time. Rather than trying to analyse the relations of all individuals in a population, the coalescent describes the (unknown) genealogy of a sample in terms of a stochastic process. If one starts with a sample of size n and traces back the ancestral history of these n lineages, one can compute the probability that at a time t two lineages in the genealogy coalesce, i.e., the most recent common ancestor (mrca) of the corresponding individuals is found. The probability depends on the sample size and the population trajectory. After a coalescent event occurs, the number of lineages is reduced by 1. The coalescent process stops when the mrca of the whole sample is found. In the situation of a stationary population of constant size, simple formulae are available that describe branch lengths in a genealogy of a sample, total length of a genealogy, etc. (11,18). If one drops the assumption of constant size, it is more difficult to find analytical solutions, whereas it is still possible to get instructive results using simulation techniques (19–22). The following simulation method to analyse PCR data bears similarity with the coalescent approach. In PCR the offspring of the initial molecules are related by a randomly growing tree (8). Instead of generating this tree independently for each initial template, we adopt the following approach. For each initial molecule the number of molecules in the PCR product in each cycle (the size trajectory) is computed (step 1 in the algorithm). Thereby, we distinguish two types of molecules: those that are immediate copies from a molecule of a previous cycle (filled circles in Fig. 1) and those that existed in the previous cycle (open circles in Fig. 1). From all molecules at the end of the PCR a random sample of n sequences is drawn. Then, we randomly assign to each of the sampled sequences one of the initial molecules as ancestor (step 2) and regard the sets of sampled sequences that are descendents of the same initial molecule as subsamples. In the next step (step 3) we trace back the genealogies of all subsamples, separately. Figure 1 illustrates this process for one initial molecule (and one subsample) according to the example in Table 2. In this example we assume that a subsample of six sequences was drawn from a total of 16 sequences. In order to generate the subsample genealogy the special features of PCR must be taken into account. A coalescent event, i.e., the merging of two molecules while going from cycle 5 to cycle 4, is only possible if exactly one of the two molecules is an immediate copy. Among the six sampled sequences three were copied during cycle 5. Hence, at most three coalescent events are possible, and in fact one such event occurred. The coalescent process stops when only one molecule is left. If only one molecule is present and the cycle number is not equal to zero, we have to determine how many replications took place from the initial molecule to this molecule. After all subsample genealogies are generated, they are combined to one single genealogy (step 4). Finally, we superimpose a mutational process on the genealogy (step 5), where mutations are only allowed where replications took place in the genealogy (thick lines in Fig. 1). Before we describe the simulation more formally we assign a number k, k = 1, …., S0 to each of the S0 initial molecules. Additional notation is summarized in Table 1. We assume that C, n, S0 and λ1, …., λC either are known or can be estimated reliably. Figure 1 View largeDownload slide Graphical illustration of a subsample genealogy according to the example in Table 2. Filled circles represent molecules that were newly amplified in a cycle, open circles represent molecules already present in the previous cycle. + indicates that the molecule is in the sample; thick lines represent a replication in the genealogy. Figure 1 View largeDownload slide Graphical illustration of a subsample genealogy according to the example in Table 2. Filled circles represent molecules that were newly amplified in a cycle, open circles represent molecules already present in the previous cycle. + indicates that the molecule is in the sample; thick lines represent a replication in the genealogy. The simulation steps are the following: Forward step (Replication step) For each initial molecule k, k = 1, …., S0, we generate the size trajectory   where i = 1, …., C. Thus Sampling step We determine how many molecules in a sample of size n stem from an initial molecule k, k = 1, …., S0. In principle, this implies sampling from a multivariate hypergeometric distribution, which is approximated by a multinomial distribution. Thus,   with . Backward step (generating subsample genealogies) For each initial molecule k, k = 1, …., S0, that has at least one descendant in the sample, i.e., NC, k > 0, we trace back the genealogy of a subsample of NC, k molecules given the trajectory {SC, k, SC−1,k, …., S1, k, S0,k = 1}, generated in step 1. a) There are Ni, k molecules in the genealogy of the subsample present after cycle i. As we go back from cycle i to cycle i − 1 we need to compute Ri, k, the number of molecules among the Ni, k molecules that were newly synthesized. Among the Si, k molecules there is a total of Si, k − Si−1,k of newly synthesized molecules in cycle i. Ri, k is equal to the number of replications in the genealogy of the subsample in cycle i and follows a hypergeometric distribution with parameters Ni, k, Si, k − Si−1,k and Si, k. Hence,   b) In the next step we generate Li, k = Ni, k − Ni−1,k, the number of coalescent events, i.e., the number of lost lineages in the genealogy of the subsample in cycle i. A lineage is lost if a template molecule and its direct descendent are both present in the genealogy of the subsample in cycle i. To determine the number of lost lineages we have to decide for each of the Ri, k newly synthesized molecules whether its template molecule is among the remaining Ni, k − Ri, k molecules of the subsample. Conditional on Ri, k, Ni, k and Si−1,k, Li, k is again a random variable with a hypergeometric distribution and parameters Ri, k, Ni, k − Ri, k and Si−1,k:   The number of molecules present in the genealogy of the subsample after cycle i − 1, Ni−1,k, is then given by:   If Ni, k = 1 for any i ≥ 1, the process reduces to a sequence of Bernoulli trials with parameters 1 − (Sj−1,k)/(Sj, k), j = i, …., 1. This simplification is possible, since, with one lineage remaining in the genealogy of the subsample, we just have to determine in how many of the first i cycles the molecule was newly synthesized. Tracing back the genealogy of the subsample ends at template molecule k (N0,k = 1). Generating one genealogy Finally, the genealogies of all subsamples, i.e. all initial molecules that have at least one descendant in the sample, are generated. Because we assume that all the initial molecules are identical we create a single genealogy by fusing all the initial molecules to one single root of the final genealogy. Mutation step Given the final genealogy we superimpose a nucleotide substitution process. For each replication in the genealogy (i.e. thick lines in Fig. 1) the number of mutations introduced by this replication is a Poisson distributed random variable with parameter μ, where m is the error rate of PCR per target sequence and per replication. Figure 2 View largeDownload slide Example of simulated probability distributions Pr(Mn = m|μ) for 100 equidistant values of μ (S0 = 100, other parameter values as described in the Applications section). Figure 2 View largeDownload slide Example of simulated probability distributions Pr(Mn = m|μ) for 100 equidistant values of μ (S0 = 100, other parameter values as described in the Applications section). Analysis step The mutation step produces a set of n not necessarily identical sequences generated under our model of PCR. We compute our favourite statistic from this data set. As an example, we compute Mn, the number of variable positions in a set of n sequences. In consideration of the two applications we introduce in the next section, we generate the probability distributions of Mn, given μ, for the range of the parameter μ we are interested in. The parameters for the simulation were adopted from a published data set (6) described in the application section. In order to compute the distribution of Mn we generated 106 genealogies. According to each of these genealogies we produced data sets of size n = 28 for 100 equidistant values of μ in the interval [0.019, 0.079] and computed the number of variable positions for each data set. The relative frequency of observing m variable positions in the simulations with a fixed parameter μ represents the probability Pr(M28 = m|μ). Figure 2 shows the result of a simulation. The distributions are smooth and unimodal. With increasing parameter μ the peak of the distribution is shifted to larger values of m and the distributions get broader. A usual simulation run with 100 × 106 generated data sets takes up to 10 h on a SUN sparc station 20, but then we are confident that the simulated distributions are good approximations of the theoretical probability distributions. Applications This section gives two examples that apply the simulation method. One question is how many errors PCR introduces into the target sequence. This is answered by maximum likelihood estimation of the error rate in a PCR. Another question is whether the initial template molecules were all identical or whether there is reason to believe that the template was heterogeneous (due to contamination, heteroplasmy of the source organism or damage in the sequences). This is answered with the use of a statistical test. Estimating μ We apply our method to a published data set. Saiki et al. (6) amplified a 239 bp region of genomic DNA. After C = 30 PCR cycles, M28 = 17 variable positions were observed when they sequenced from n = 28 different clones, Furthermore, the authors measured the extent of amplification after 20, 25 and 30 cycles. They report an increase of 2.8 × 105, 4.5 × 106 and 8.9 × 106 respectively. This corresponds to an overall efficiency of 0.705 in 30 cycles (6). We also determined cycle specific efficiencies from the data using the following formula:   From the reported increase after 20, 25 and 30 cycles we computed:   These values for λi were used in the simulations. Since no information about the number of initial molecules is given, the analysis was carried out for different S0 values (1, 10, 100, 1000). We generated probability distributions Pr(Mn = m|μ) for 100 equidistant values of μ in the interval [0.019, 0.079]. If one takes the observed number of variable positions in the sample and cuts along this line through Figure 2 one gets lik(μ|M28 = 17), the likelihood function of μ given M28 = 17. Figure 3 shows the likelihood functions for our choices of S0. For each S0 the position of the maximum of the likelihood function is the maximum likelihood estimate. We use Fisher information to estimate the standard error of the estimate. In addition to our estimation results Table 3 shows the method of moments estimate from Sun (7) assuming constant efficiency λ = 0.705. From Table 3 as well as from Figure 3 one readily sees that the estimate converges for increasing value of S0. This is due to the fact that with increasing S0 the probability that each of the sampled molecules stems from a different initial molecule also increases. In this case the sampled molecules represent an independent sample from a PCR. The impact of the varying efficiency (that cannot be incorporated into the method of moments estimation) is the reason for the deviation of the two estimation methods. If one assumes the same (constant) efficiency, the ML-estimate converges to the method of moments estimate for increasing S0. Table 3 View largeDownload slide Estimation of μ (error rate per target sequence and replication) for different S0 from a published data set (6) together with estimated standard errors Table 3 View largeDownload slide Estimation of μ (error rate per target sequence and replication) for different S0 from a published data set (6) together with estimated standard errors Figure 3 View largeDownload slide Simulated likelihood functions lik(μ|M28 = 17) for a published data set (6). The numbers in the graph represent the used S0 values (other parameter values as described in the Applications section). Figure 3 View largeDownload slide Simulated likelihood functions lik(μ|M28 = 17) for a published data set (6). The numbers in the graph represent the used S0 values (other parameter values as described in the Applications section). A test for homogeneity In some applications of PCR, particularly in forensic medicine, it is very important that one can be sure that the result is not flawed by contamination of the template. In contrast, when heteroplasmy of a DNA sequence in an individual is suspected, one would like to confirm that the observed differences in the sample are not due to replication errors in PCR. In both situations the following statistical test can be applied. The hypothesis H0: the initial template molecules are homogeneous is tested. Since we use the number of variable sites in a sample as a test statistic, we can translate the hypothesis to H0: all variable positions observed in a sample are replication errors introduced by PCR. If we have a reliable estimate of the mutation rate from a pre-study using the same experimental conditions, we can readily use the simulations to generate and derive the test. Given the simulation parameters, and the observed number of variable positions, the simulation procedure returns , the P-value of the observation m. If the P-value is smaller than a given significance level α, H0 would be rejected. In order to investigate the power of this test we used the data set of Saiki et al. (6) together with the ML-estimates from above. The test has reasonable power, as illustrated by Figure 4. The horizontal axis represents the hypothesis space in terms of variable positions that are introduced by the heterogeneity of the initial molecules and not by PCR. No variable position represents H0. The vertical axis shows the probability of rejecting H0 as a function of the variable positions already present among the initial molecules. If we do not have a reliable estimate of the error rate, then the function f(μ) = Pr(Mn ≥ m|μ), the P-value of the observation m as a function of μ can be deduced from the simulation. We chose a significance level α. For all values of μ with f(μ) 3 α the observed data would lead to a rejection of H0. In particular in forensic studies, where sometimes only very few molecules are available for PCR, the influence of S0 is an important parameter for f(μ). Figure 5 illustrates this situation with α = 0.05 and m = 17. Depending on S0 the function f(μ) intersects the horizontal line α = 0.05 at different values of μ. The mutation rate where f(μ) = 0.05 is called μmax. If the true value μ is less than μmax, then Pr(Mn ≥ m|μ) < 0.05 and H0 is rejected. Figure 4 View largeDownload slide Simulated power curve of the homogeneity test (S0 = 100, μ = 0.051) shows the probability to reject H0 as a function of the number of variable positions (vp) that are introduced by the heterogeneity of the initial molecules. No variable position represents H0. Figure 4 View largeDownload slide Simulated power curve of the homogeneity test (S0 = 100, μ = 0.051) shows the probability to reject H0 as a function of the number of variable positions (vp) that are introduced by the heterogeneity of the initial molecules. No variable position represents H0. Figure 5 View largeDownload slide Simulated f(μ) = Pr(M28≥17|μ), the P-values of the observation M28 = 17 as a function of μ for a published data set (6). The horizontal line represents a significance level of α = 0.05. The numbers in the graph represent the used S0 values (other parameter values as described in the Applications section). Figure 5 View largeDownload slide Simulated f(μ) = Pr(M28≥17|μ), the P-values of the observation M28 = 17 as a function of μ for a published data set (6). The horizontal line represents a significance level of α = 0.05. The numbers in the graph represent the used S0 values (other parameter values as described in the Applications section). In addition the maximum likelihood estimate present in the previous section yields the Wald test for μ. This test is based on the asymptotic normality of the MLE and uses the standard error based on Fisher information. According to the Wald test, we can reject as too small any value of μ less than (if α = 0.05). The Wald test using the estimates from Table 3 leads to slightly more conservative results for μmax, but relies strongly on an accurate estimation of . Table 4 summarizes the values of μmax for both methods and different S0. If a rough estimate of μ is available from similar studies (or from literature) that is clearly smaller (or larger) than μmax, we can reject (or confirm) the null hypothesis of homogeneous initial template molecules. Thus, estimates of the efficiencies λi and an idea of μ leads to a simple procedure to decide if the template was homogeneous or not. Table 4 View largeDownload slide Maximum error rates that lead to rejection of the null hypothesis of homogeneous initial template molecules for different S0 and α = 0.05 [(sim), (Wald) see text] Table 4 View largeDownload slide Maximum error rates that lead to rejection of the null hypothesis of homogeneous initial template molecules for different S0 and α = 0.05 [(sim), (Wald) see text] Discussion The coalescent approach combined with an efficient simulation algorithm is the key to the analysis of DNA sequence samples drawn from a PCR product. We were able to generalize the replication process model by allowing cycle dependent efficiency and explicitly incorporating the number of initial template molecules. This model is more realistic, since it is well known that synthesis of new sequences depends on the concentration of total PCR product and reagents in the PCR tube (6,23), which are changing from cycle to cycle. In this article we used an infinite site model for the mutation process. The simulation method is not restricted to this. In principle it is possible to implement any other mutation process model [e.g. Tamura and Nei's model (24)] into the algorithm. Then, however, estimation of the relevant parameters gets increasingly difficult. One application of the simulation procedure is the maximum likelihood estimation of the error rate in PCR under a general replication process model. Compared to other methods (3–9) our procedure makes fewer assumptions and shows a comparable precision (measured in terms of standard errors). The test for homogeneity of the template is a new application that uses the information in a set of sequences from a PCR. To apply the test some pre-studies are necessary. Quantification experiments have to be done in order to get estimates of the efficiency under various PCR conditions and template concentrations. These controlled experiments are the basis of PCR error rate estimation. However, once the properties of PCR are well understood and reliable estimates of the error rate and the amplification behaviour are available in standard situations, the test will be a powerful tool to detect contamination, heteroplasmy or DNA damage. This will have an impact on forensic medicine and other scientific fields. Acknowledgements Part of the work was accomplished while one of us (G.W.) was visiting Simon Tavaré at the University of Southern California, Los Angeles. The authors are grateful to S.Tavaré and P.Marjoram for fruitful discussions and suggestion of a coalescent approach. We also wish to thank Ellen Baake and two anonymous reviewers for helpful comments and for improving the manuscript. This work was supported by a grant from the DFG to A.v.H. (Ha1628/2-1). The program mlpcr can be retrieved free of charge over the Internet from URL: http://www.zi.biologie.uni-muenchen.de/~gweiss/mlpcr.html. 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TI - A coalescent approach to the polymerase chain reaction
JO - Nucleic Acids Research
DO - 10.1093/nar/25.15.3082
DA - 1997-08-01
UR - https://www.deepdyve.com/lp/oxford-university-press/a-coalescent-approach-to-the-polymerase-chain-reaction-AKrRPa2T2e
SP - 3082
EP - 3087
VL - 25
IS - 15
DP - DeepDyve
ER -