Analysis of enzyme kinetic data for mtDNA replication

Abstract

A significant amount of experimental data on the reaction kinetics for the mitochondrial DNA polymerase gamma exist, but interpreting that data is difficult due to the complex nature of the function of the polymerase. In order to model how these measured kinetics values for polymerase gamma affect the final function of the polymerase, the replication of an entire strand of mitochondrial DNA, we implement a stochastic simulation of the series of reaction events that the polymerase carries out. These reactions include the correct and incorrect polymerization events, exonuclease events which may remove both incorrectly and correctly matched base pairs, and the disassociation of the polymerase from the mitochondrial DNA template. We also describe other reactions which may be included, such as the addition of nucleoside analog tri-phosphates as substrates. The simulation analysis of the kinetics data is implemented through a standard Gillespie algorithm. We describe the methods necessary to define, code and test this algorithm, as well as describing the hardware and software options that are available.

Introduction

Mitochondrial polymerase gamma is the sole DNA polymerase active in mitochondria [1] and is responsible for the replication of mitochondrial DNA (mtDNA). Vertebrate polymerase gamma is composed of two subunits: a catalytic core, Pol γ–α, that contains the DNA polymerase and 3′–5′ exonuclease activities, and an accessory subunit, Pol γ–β, which enhances catalytic activity and serves as a processivity factor during DNA synthesis [2].

The function of polymerase gamma is far more complicated than the function of a typical enzyme which converts a substrate into a product. For that reason, the analysis of the enzyme kinetics of polymerase gamma must also be far more complicated. The function of polymerase gamma is the replication of a complete mitochondrial DNA molecule, while the enzyme kinetics data for polymerase gamma are at the level of individual reactions at the base-pair level. One tool that can be used to bridge that large gap between the enzyme kinetics data and the final product of a replicated DNA molecule is simulation. A computational simulation can reproduce each individual reaction of the polymerase activity, at the lowest level for which we have experimental data on the relevant reaction kinetics of the polymerase. The simulation can tirelessly repeat this process for the several tens of thousands of reactions necessary for the replication of a single strand of human mtDNA. Since mutations introduced through polymerase errors are of great practical interest, we need to use a stochastic simulation approach that will allow such replication errors to occur with probabilities that are calculated from the experimentally measured enzyme kinetics.

The “Gillespie algorithm” (or “Stochastic Simulation Algorithm”) is a well-known Monte Carlo simulation method for chemical reactions [3]. This algorithm takes a defined list of possible reactions and uses the reaction rates, based on the measured enzyme kinetics data and the substrate concentrations, to calculate the probability of each reaction on the list. These probabilities are then used to randomly choose the sequence of reactions which occur. The original Gillespie algorithm is particularly useful for simulating reactions involving a small number of molecules, while more elaborate and approximate versions of the basic algorithm have been created in order to handle large numbers of molecules [4].

Traditional continuous and deterministic biochemical rate equations are modeled as a set of coupled ordinary differential equations but these methods rely on bulk reactions that require the interactions of millions of molecules. In contrast, the Gillespie algorithm allows a discrete and stochastic simulation of a system with few reactants because every individual reaction is explicitly simulated. The physical basis of the Gillespie algorithm is the collision of molecules within a reaction vessel where the reaction environment is assumed to be well mixed. The general Gillespie algorithm can be summarized by the following series of steps:

Step 1, Initialization: Initialize the number of molecules in the system, the reaction kinetics constants, and the random number generators.

Step 2, Calculate reaction probabilities from reaction rates: Based on the substrate concentrations and the enzyme kinetics data, reaction rates R_i are calculated for each possible reaction, where i is an index denoting the particular reaction from the reaction list. The probability P_i for each reaction is then calculated from the list of n reaction rates through the following equation:

$$P_i=R_i/R_{\mathit{total}}$$

The parameter R_total is the sum of all of the reaction rates, as follows:

$$R_{\mathit{total}}=\sum _{j=1}^n{R}_j$$

Note that the sum of the probabilities is 1, with this definition.

Step 3, Choose a reaction: Use a pseudo-random number generator to generate a uniform random number r_reaction in the range [1] and use this random number to choose the one reaction that occurs from the list of n possible reactions. The reaction number j is chosen if it satisfies the following condition:

$$\sum _{i=1}^{j-1}{P}_i<r_{\mathit{reaction}}⩽\sum _{i=1}^j{P}_i$$

Then a second uniform random number r_time, also in the range [0,1] is chosen. This random number is used to set the time τ required for this reaction by the following equation:

$$\tau =-(1/R_{\mathit{total}})\ln (r_{\mathit{time}})$$

It is important to note here that the sum of all of the reaction rates is used in defining the time τ, not just the rate of the one particular reaction that was chosen.

Step 4, Update the variables: Increase the time by τ. Update the molecule count based on the reaction that occurred.

Step 5, Iterate the process: Check to see if any stopping condition has been met. If not, return to step 2 to recalculate the reaction rates, which may have changed due to the reaction chosen and carried out in steps 3–4.

These five steps define the basic Gillespie method for stochastic simulation. For applications of the stochastic simulation that involve large numbers of interacting molecules this basic method is inefficient in many ways, though it is an exact solution of the relevant stochastic master equation. In those cases more advanced and efficient methods should be used [4]. However, for modeling polymerase gamma activity we are generally concerned with only a single enzyme molecule, the one molecule of polymerase gamma that is replicating a particular strand of mtDNA. In this special case, the simple Gillespie algorithm outlined above is the preferred simulation method.