Team:Hong Kong HKUST/Model/Recombination
Team Hong Kong - HKUST
Single Cell ODE Model for the Recombination Module
1 Overview
The final module in our construct is the Recombination module. In this module, we use Cre recombinase to separate the origin of replication from the rest of the plasmid. Plasmids whose origin of replication has been recombined out are referred to as ”knocked out” plasmids.
In our model, we seek to model the knockout efficiency after some set amount of time has elapsed since the introduction
of Cre recombinase into the cell culture, which we define as . That is, it
is the proportion of plasmids whose origin of replication is knocked out from the original set of plasmids. Here, we
start the simulation with Cre being constitutively produced from the Time Delay module at the same concentration.
The system of reactions generally follow the analysis of Cre recombinase in [1] and can be visualized as in the diagram below.
In the construct, there are two lox sites on the plasmid which flank the origin of replication. When both lox sites are bound to two Cre molecules, the plasmid (S) can undergo recombination where the plasmid is cut and separates into two plasmids (P1, P2), in which one has the origin of replication and the other does not.
As shown in the diagram above, the binding of Cre to the lox sites can be in different orders. Notably, the binding of Cre to molecules to a lox site is cooperative, meaning that the binding of the second Cre to a lox site is made more favorable when the first Cre has already bound to that site.
Once there are four Cre molecules on the same plasmid, the Cre recombinase can catalyze a recombination reaction to produce a recombined product (P). The reverse reaction also occurs.
This product can separate into two separate plasmids (P1, P2) whose Cre can bind and unbind from the lox sites in the same manner as before
The knockout efficiency can be calculated as .
2 Approximations and Simplifications
2.1 Recombination Simplification
In our model, we do not consider the other products that could form from Cre-lox recombination. In particular, two original plasmids with 2 Cre at one lox site on each plasmid could theoretically recombine to make a joint plasmid together based on the principle of Cre recombinase. However, this is less likely because the lox sites are located much further apart than those on the same plasmid. Thus, in order to simplify our model and limit the number of species to be considered, we do not consider them.
3 System of Reactions
We can describe the Recombination module as elaborated above with this set of reactions:
4 Simulations
To simulate the system, we implemented Euler’s method for approximating ordinary differential equations and applied the approximations as stated.
4.1 Cre-lox Recombination
Fig. 1 Simulation of Cre Recombinase kinetics
In this simulation, it can be seen that the knockout efficiency of Cre tends towards 53.5 % after around 5000 seconds (around 83 minutes). While this knockout efficiency is not particularly high, we will show in the population dynamics model that, over time, it results in most of the plasmid eventually being those of the knocked out.
4.2 Cre-lox Recombination without Reverse Reaction
In our construct we use lox66 and lox71, which are mutants of loxp for which the forward reaction is highly preferred over the reverse reaction because the product of recombination includes a mutant loxp which has a very low tendency to recombine. Because we were unable to find reaction constants for the reverse reactions of lox66 and lox71 in the literature, we simply simulated a version of the model where the reverse reaction is ignored for reference.
Fig. 2 Simulation of Cre Recombinase kinetics without reverse reaction
In this simulation, it can be seen that the knockout efficiency of Cre is around 95% after around 5000 seconds (around 83 minutes). Without the reverse reaction, this would eventually approach 100% over time. However, it should be noted that our model deliberately ignores the reverse reaction with may occur in reality even with lox66 and lox71.
5 Model Simplification
In this simulation, due to the high values of reaction constants, we needed to use a very small time step in the simulation which led to a long simulation time. In order to make it feasible to simulate further models, we will attempt to fit the progress of the module with a simple system of differential equations.
Particularly, because the rate of the forward reaction is only dependent on the amount of unrecombined plasmid in the system, and the reverse reaction is only dependent on the amount of recombined plasmid, we will simplify the system to have only S and P where S is now the proportion of unrecombined plasmid and P is the recombined fraction. We thus attempted to fit it into a differential equation of the form
which would correspond to having the reaction system
We fitted the values of r+ and r− using MATLAB, which resulted in the following graph for S and P.
Fig. 3 Comparison of fitted and simulated kinetics
As can be seen, the fitted values of r+ and r− result in a curve that follows the results from the simulation of the actual kinetic system quite closely.
6 Parameters
| Parameter | Value | Justification | Description |
|---|---|---|---|
| S | 1.00E-06 | Estimated | Initial concentration of S (Original Construct) |
| Cre | 1.00E-04 M | Simulated concentration from Time Delay model | Constitutively produced concentration of Cre Recombinase |
| [S-Cre1]0 | 0 M | Controlled Variable | Initial concentration of S-Cre1 |
| [S-Cre2A]0 | 0 M | Controlled Variable | Initial concentration of S-Cre2A |
| [S-Cre2B]0 | 0 M | Controlled Variable | Initial concentration of S-Cre2B |
| [S-Cre3]0 | 0 M | Controlled Variable | Initial concentration of S-Cre3 |
| [S-Cre4]0 | 0 M | Controlled Variable | Initial concentration of S-Cre4 |
| [P] | 0 M | Controlled Variable | Initial concentration of P |
| [P1-Cre2]0 | 0 M | Controlled Variable | Initial concentration of P1-Cre2 |
| [P2-Cre2]0 | 0 M | Controlled Variable | Initial concentration of P2-Cre2 |
| [P1-Cre1]0 | 0 M | Controlled Variable | Initial concentration of P1-Cre1 |
| [P2-Cre1]0 | 0 M | Controlled Variable | Initial concentration of P2-Cre1 |
| [P1]0 | 0 M | Controlled Variable | Initial concentration of P1 |
| [P2]0 | 0 M | Controlled Variable | Initial concentration of P2 |
| k1 | 2.20E+08 /M/s | [4] | Reaction constant |
| k2A | 2.20E+08 /M/s | [4] | Reaction constant |
| k2B | 2.30E+08 /M/s | [4] | Reaction constant |
| k3A | 2.30E+08 /M/s | [4] | Reaction constant |
| k3B | 2.20E+08 /M/s | [4] | Reaction constant |
| k4 | 2.30E+08 /M/s | [4] | Reaction constant |
| k5 | 6.00E-03 /s | [4] | Reaction constant |
| k6 | 3.30E-04 M/s | [4] | Reaction constant |
| k-1 | 6.60E-02 M/s | [4] | Reaction constant |
| k-2A | 6.60E-02 M/s | [4] | Reaction constant |
| k-2B | 4.80E-03 M/s | [4] | Reaction constant |
| k-3A | 4.80E-03 M/s | [4] | Reaction constant |
| k-3B | 6.60E-02 M/s | [4] | Reaction constant |
| k-4 | 4.80E-03 M/s | [4] | Reaction constant |
| k-5 | 5.20E-03 /s | [4] | Reaction constant |
| k-6 | 8.30E+07 /M/s | [4] | Reaction constant |
| r+ | 6.00E-04 /s | Fitted with MATLAB | Fitted forward reaction constant |
| r- | 5.19E-04 /s | Fitted with MATLAB | Fitted reverse reaction constant |
