Errors induced during PCR amplication

The Polymerase Chain Reaction (PCR) is one of the most widely used techniques in modern molecular biology to amplify a single or few copies of a piece of DNA sequences. The PCR products are used as inputs of many other applications such as diagnosis of diseases. Therefore the accuracy of the further analysis depends on the quality of the PCR outputs. In order to study the amplification process in PCR more carefully, the mathematical modelling is essential. Since 1989 different models are presented in the literature. In the perfect PCR all the replicated molecules are the exact copies of the original ones and the number of molecules is doubled after each replication cycle. But in practice these two conditions cannot be fulfilled. All the present mathematical models consider dissatisfaction of at least one of these conditions. The first condition says that all the replicated sequences should be exactly the same. But in the real PCR in addition to the desired sequences, some artificial molecules are produced. These artificial sequences can be the results of mutation or they are generated because of undesired reactions between different template molecules, These artificial molecules are called “Chimera”and “Heteroduplex”in the literature. The fraction of the mutant sequences can be calculated by the distribution of number of mutations. In this thesis mutation is simulated as a stochastic process to get a distribution of number of mutations. The results show that the ratio of mutants depends on the multiplication of the sequence length and mutation rate. The mutation rate is determined by the performance of the DNA polymerase. So using an accurate polymerase, about 98% of the sequences are amplified without mutation when they are not so long (e.g. 100 bp). But the probability to have mutants is higher for longer sequences (e.g. 1000 bp). The results show that about 12% of the generated sequences have one mutation even by using an accurate enzyme. Mutation may also change the pairwise Hamming distances between sequences when the PCR inputs are mixture of different genes. Mutation may cause the pairwise distances between different genes decrease or increase or even remain constant. When the sequences are very close to each other and the Hamming distances change a lot, it is not easily possible to detect the original genes by comparing their pairwise distances. So the changes of the pairwise Hamming distances during replication cycles are also simulated and the distribution of the pairwise Hamming distances is presented for different normalized Hamming distances between sequences. The results show that when the sequences are very distant it is more likely that a mutation happens but the pairwise distance does not change, but for closer sequences a mutation usually changes the Hamming distance. They also imply that the sequences get further when they are close to each other (normalized distances of 5% and 10%) and get closer when they are very distant (normalized distances of 90% and 95%). The second condition is referred as a PCR efficiency, it is determined in each cycle by the proportion of the sequences which make a copy from themselves to the total sequences in that cycle. So the PCR efficiency will be 100% if all sequences make a copy from themselves. The efficiency is almost constant in the first replication cycles. But as the number of molecules grows exponentially, the environment is fully condensed by the DNA sequences and a smaller proportion of them participate in the replication process. So the efficiency or amplification rate decreases druring PCR cycles. When the PCR input sequences are mixture of different genes with different ratios the efficiency does not decrease with the same rate for all of them. This problem has been addressed in the literature as a result of the template re-annelaing. Different amplification rates cause a bias towards amplifiying some genes more than the others. Some other reasons are also reported which give such bias such as GC content and primer mismatch. In this thesis a mathematical model which describes template re-annealing is simulated. The results are given for some cases based on the experimental data. They show that how the genes ratios change during the PCR cycles and how the amplification rates decrease because of the template re-annealing. The simulation is run for different conditions to study the effective parameter and to find a proper PCR setting to remove this bias.