Team:NUS Singapore-Sci/enzyme kinetics

NUS Singapore Science: InterLab

Enzyme
Kinetics

Enzyme Kinetics and Least Squares Regression
The efficiency of our RESCUE system is likely to be dependent on multiple factors such as mismatch distance, length of spacer regions as with ADAR-dCas13b constructs (Cox et al., 2016), as well as the relative concentrations of the substrates/enzymes. The concept of regression models can be utilized to identify and evaluate the significance of these factors from experimental results. As such, a early build of an enzyme kinetics regression model on dCas13b-APOBEC editing efficiency may help us to gain further insights of the RESCUE system.
Goal
  1. 1. Simulate the RESCUE system under different relative concentrations of substrates and enzymes to determine the concentrations that might yield maximum efficiency.
  2. 2. Determine the change in the binding and catalytic efficiency when spacer length and mismatch distance is varied, which will help in the design of gRNA for more efficient base editing.
Our assumptions of the model are as follows:
  1. 1. Association between dCas13b and gRNA is reversible and precedes enzyme-gRNA complex association with substrate mRNA. This is because dCas13b requires the gRNA to bind to the correct target sequence.
  2. 2. Once the Enzyme-gRNA-Substrate-mRNA trinity complex (ERS) is formed, the reaction will proceed in a single direction to produce the cleaved product.
First, we build upon a kinetics model for our fusion protein product.


Figure 1: Overall biochemical kinetics equation of the RESCUE system. The coefficients (parameters) of the various reaction rates are denoted as k­I, where i represents the differing stages. The growth rates of each biological substrates are denoted by Gi, and while the decay rates are denoted as Di. In particular, K2/k-2 denotes the binding efficiency (dissociation constant) and k3 denotes catalytic efficiency.
Where:
S - Unedited Substrate mRNA
R - Guide RNA
E - Enzyme for C to U editing
ER - Enzyme-gRNA complex
ERS - Enzyme-gRNA-Substrate complex
P - Edited Product mRNA
F - Green Fluorescent protein produced

Given the set of kinetic equation as defined above, the rate of change of each protein/RNA can then be described in the following differential equations as follow:

$$\delta_{t}S = G_{s} - D_{s\cdot}S -k_{2}\cdot ER\cdot S +k{-2}\cdot ERS \quad \textrm{(Equation 1)}$$
$$\delta_{t}ER = k_{1}\cdot E\cdot R - k{-1}\cdot ER - k_{2}\cdot ERS +k_{-2}\cdot ERS \quad \textrm{(Equation 2)}$$
$$\delta_{t}ERS = k_{2}\cdot ER\cdot S - k_{3}\cdot ERS - k_{-2}\cdot ERS \quad \textrm{(Equation 3)} $$
$$\delta_{t}P = k_{3}\cdot ERS - D_{s}\cdot P \quad \textrm{(Equation 4)} $$
$$\delta_{t}F = k_{4}\cdot P - D_{f}\cdot F \quad \textrm{(Equation 5)} $$

where Equation 4 assumes that ERS to ER + P is an non-energetically favourable step, and hence is unlikely to occur.

The differential equations, when plotted, yield a complex, non-linear kinetic curve that can be comparable to the standard Komod kinetics curve. But unlike Komod kinetics, logarithmic conversion of the axes may not always result in a linear plot. Non-linear least-square regression (LSR) is thus an alternative strategy that can be used to obtain a best fit curve for enzymatic assays, capable of generating a set of possible values for the coefficients within the differential equations.


Figure 2: Least-squared regression started off with a set of initial coefficients (parameters), provided by the user. (Left) The parameters are updated, through increasing and decreasing the different parameters by a small value until the parameter set that provides the lowest squared differences is obtained. (Right) A 2D ball on a slope analogy to the gradient descent method used for the least-squared regression. Ball will continue to travel downslope until the minimum point (minima). Gradient descent method is used to determine the lowest sum of squared differences value (local minima), which can indicate correct readings for the various parameters.
The sum of squared differences is the sum of the computed y-axis differences of each readings to their respective points on the current curve. Mathematically, the squared differences (abbreviated as D) can be expressed through Equation 6 below.

$$D = \sum_{i=1}^{n} (y_{i}-p_{i})^2 \quad \textrm{(Equation 6)} $$