Team:Hong Kong HKUST/Model/Population
Team Hong Kong - HKUST
Population Dynamics
1 Overview
In this section, we want to model the proportion of the plasmid that is knocked out after the addition of Cre when accounting for population dynamics, namely cell replication and death. Here, as before, we wish to find how the gene of interest (GOI) concentration will eventually decrease. Here, we assume that the population has reached carrying capacity and so the rate of cell replication is equal to the rate of cell death. We also assume all cells with the construct have Cre constitutively produced after the activation of the construct.
As described in the Cre module model, the GOI is separated from the origin of replication (ORI) when Cre recombinase cuts the construct at the lox sites.
During each cell replication, all plasmids with an ORI will be replicated before the cell divides. If an unrecombined construct (with both the GOI and the ORI) replicates, it would create another copy of itself. Any plasmids with only the GOI cannot replicate as they lack an ORI. The plasmids with only an ORI would also replicate.
Here, we will refer to the proportion of unrecombined plasmid as S, the proportion of the recombined version (having the GOI and ORI in separate plasmids) as P, and the proportion of ORI-only plasmids from the replication of the recombined version as N.
Referring to the figure below, during cell replication, when S replicates, it will generate a copy of S, while when P and N replicate, they will generate a copy of N. Because we assume the cell population is at the carrying capacity, meaning that the rate of cell replication will be equal to the rate of cell death. Thus, we will only have to consider the replication and death mechanism for P because the rates are balanced for S and N.
Because we could not confirm the reaction constants for lox66 and lox71, we decided to proceed with the model using the simplified reaction kinetics for lox.
2 Approximations and Simplifications
2.1 Recombination Simplification
In this model, we consider P separately from N as a different entity. That is, P is a GOI plasmid and an ORI plasmid, while N is an ORI plasmid that does not have a corresponding GOI copy. We make this simplification so that we may use the fitted model from the Cre module.
2.2 Constant Cre Recombination Rate Approximation
In this model, we assume that Cre is constitutively produced, and that the forward and reverse reaction constants remain constant. In reality, the concentration may have changes, such as during replication, but because the amount of time spent under such conditions should be short, we decide to simplify the model by not considering them.
3 System of Reactions
We can describe the model as elaborated above with this set of reactions:
The first two reactions come from considering the death and replication of P respectively, while the second two come from the fitted result of the Cre model.
4 Simulations
To simulate the system, we implemented Euler’s method for approximating ordinary differential equations and applied the approximations as stated.
Fig. 1 Simulation of knockout efficiency with cell death and replication
As we can see from the graph, the concentration of both S and P gradually approach 0 over time.
This is because the replication of P only produces N while cell death would reduce P. As P decreases, the reaction equilibrium between S and P is pushed in the forward direction in accordance with Le Chatlier’s principle, and so S will also decrease. Meanwhile, N’s proportion is being increased do to the replication of P and eventually all of the plasmids are in the form of N.
This model suggests that if our construct works correctly, then all of the original construct will eventually be knocked out given enough time if the cell culture is at carrying capacity.
5 Parameters
| Parameter | Value | Justification | Description |
|---|---|---|---|
| r+ | 6.00E-04 | Fitted with MATLAB | Fitted forward reaction constant from Recombination model |
| r- | 5.19E-04 | Fitted with MATLAB | Fitted reverse reaction constant from Recombination model |
| k | 8.33E-04 | Estimated from E. coli replication time | Replication and death constant |
