1. PRINC E TON School of Engineering and Applied Science Characterizing Mathematical Models for Polymerase Chain Reaction Kinetics Ifunanya Nwogbaga, Henry Mattingly, and Stanislav Y. Shvartsman Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ Abstract Polymerase Chain Reaction (PCR) is a technique to amplify a DNA sequence into thousands or millions of copies. This makes it indispensable in biotechnology and biomedical research. People use kinetic models to describe the dynamics of PCR, tackle the quantitative PCR (qPCR) problem: estimating initial DNA concentrations from measurements of DNA amplification versus cycle, and study the accuracy of these estimates. Our goal was to learn about models of PCR of varying complexities to design lecture slides for a graduate level kinetics course and to explore the qPCR problem. The first model we studied, proposed by Gevertz and others, did not use consistent state variables throughout the steps of PCR. The second model, proposed by Mehra and Hu, was more complex, complete, and consistent than Gevertz, but was not sensitive to specific DNA sequences and temperature protocol of PCR. The third model, proposed by Marimuthu and others, laid out a thermodynamic and kinetic theory for estimating temperature- and sequence-dependence rate constants to use in their model, which was very similar to Mehra and Hu’s model. After reproducing the Gevertz and Mehra models and studying the Marimuthu model, we decided that the Mehra model had the appropriate level of completeness and simplicity for teaching. After exploring the qPCR problem, we realized that the Mehra model produced seemingly more accurate estimates of initial DNA concentrations than the Gevertz model; however, the Gevertz model produced better fits to data when trying to estimate other parameters, which we found peculiar. This indicates that the difficulty of fitting kinetic models to data increases with the complexity of the model. Thus, we will study Marimuthu’s model closer in depth by seeing how well the most complicated of the three models handles the qPCR problem and continue to compare the Gevertz and Mehra models. Acknowledgements and References This research was made possible by the generous support of the Lewis-Sigler Institute for Integrative Genomics. 1.Boggy, G.J., Woolf, P.J. A Mechanistic Model of PCR for Accurate Quantification of Quantitative PCR Data, 30 August 2010. PLoS ONE 5(8): e12355. doi:10.1371/ journal.pone.0012355 2.Gevertz, J.L., et al. Mathematical Model of Real-Time PCR Kinetics, 5 November 2005. Biotechnology and Bioengineering, Vol. 92, NO. 3 3.Marimuthu, K., et al. Sequence-Dependent Biophysical Modeling of DNA Amplification, Biophysical Journal, October 2014, Vol. 107, pp. 1731-1743. 4.Marimuthu, K. and Chakrabarti, R. Sequence-Dependent Theory of Oligonucleotide Hybridization Kinetics, The Journal of Chemical Physics, 7 May 2014. 140, 175104; doi: 10.1063/1.4873585 5.Mehra, S. and Hu, W. A Kinetic Model of Quantitative Real-Time Polymerase Chain Reaction, 30 September 2005. Biotechnology and Bioengineering, Vol. 91, NO. 7 Conclusions and Future Work Results and Discussion Introduction Polymerase Chain Reaction (PCR) is a lab technique used to amplify part of a DNA sequence into thousands or millions of copies for later study. PCR is conducted in three phases: melting, annealing, and extension occurs (Figure 1). All three of these phases constitute one complete cycle of PCR and they occur at different temperatures (Figure 2); PCR is run for 25-40 cycles. Figure 1: The three main steps that occur during the amplification of a DNA sequence during each cycle of Polymerase Chain Reaction. 1 Figure 5: This figure displays the growth curve of DNA, in terms of fluorescence intensities provided by Boggy 1, for 40 cycles of PCR and the fit to the growth curve using MATLAB’s least squares solver using the state-space models designed by Gevertz 2 (top) Mehra and Hu 5 and (bottom). These phases can be written in the form of chemical reactions (Figure 3). Figure 3: Chemical equations that describe each step of PCR as given by Mehra and Hu. 5 Modeling Figure 4: The state-space model was produced by Mehra and Hu. 5 (A) (B) (C) (A) (B) (C) Figure 2: A typical temperature profile of PCR for each step. 3 These chemical reactions can be rewritten as a system of differential equations, known as a state-space model. 3,4 State-space models produced by Mehra and Gevertz were solved with MATLAB using MATLAB’s ode15s solver. After solving, we could plot the evolution of DNA concentration vs cycle number. Using MATLAB’s lsqnonlin function, which solves nonlinear least-squares curve fitting problems, we were able to fit the models to noisy data and estimate initial concentrations of DNA for a varying number of unknown parameters. We learned about some of the difficulties of recovering unknown model parameters by data fitting. To obtain estimates from the least squares solver with low uncertainty, one needs a good initial guess, a model that is sensitive to the changes in the parameter values, and enough data with a reasonable amount of noise. The more parameters there are to estimate, the harder it becomes to obtain estimates with low uncertainty (and longer it takes for the code to run). We are unsure exactly how the complexity of a model affects how well it can be used to produce a fit to data. Therefore, we plan to further analyze this phenomenon. We plan to further study Marimuthu’s model and the theory behind it to develop a simpler, more intuitive way of explaining the model in the future. Using our knowledge from studying Mehra & Hu’s model and Gevertz’s model, we have a better intuition of what type of results to expect from Marimuthu’s model. Figure 6: This figure displays the differences DNA concentration estimates that both the Gevertz 2 and the Mehra and Hu 1 models could produce for a range of initial DNA concentrations (top graph) and in a situation where all the reaction rate constants were known (data points on the left) or unknown (data points on the right); (bottom graph).