Modeling Temporal page

Why modelling plasmid dynamics?

Synthetic biology aims to engineer a living cell to perform well-defined tasks. In order to achieve these aims, biologists introduce genes into cells using self-replicating DNA carriers called plasmids. Therefore, the levels of these molecules must be tightly regulated for our system to behave correctly in a predictable fashion. Plasmid dynamics have been studied exhaustively over these years using mathematical models: from how plasmid growth is regulated to plasmid partitioning after cell division. However, little is known about how two identical plasmids interact each other.
In this part of our modelling project aims to shed light upon this issue using previous models of ColE1 plasmids and the properties when two different plasmids with same origin of replication are introduced in a cell: Do they behave independently, that is, in an orthogonal way? To address this question, we employed both deterministic and stochastic models to create a theoretical framework on this field. Also, we developed some approximations to discover some important features of the system.


Deterministic model

This model is taken from (Paulsson, 2001). Here, the plasmid levels are regulated both by a RNA molecule that inhibits plasmid replication and the dilution of plasmids as a result of cell growth and division>:

$$ \frac{d P}{dt} = r \cdot P \cdot R(s) - \frac{\log 2}{\tau} \cdot P $$ $$ \frac{d s}{dt} = \alpha \cdot P - \beta \cdot s $$ where:
$P$ is the number of plasmids
$s$ is the number of RNA inhibitors
$r$ is the plasmid growth rate
$\tau$ is the cell growth rate
$\alpha$ is the RNA inhibitor synthesis rate
$\beta$ is the RNA inhibitor decay rate
$R(s)$ is the repression effect of RNA inhibitor: $$ R(s) = \frac{1}{(1+\frac{s}{s_0})^n} $$ where $s_0$ is the value of RNA inhibitor to repress half of the replication events and $n$ is the cooperativity


Plasmids levels are regulated both by a negative feedback and cell growth. In order to take into account the presence of several plasmids, it has been modified to obtain this set of equations:

$$ \frac{d P_i}{dt} = r_i \cdot P_i \cdot R(s) - \frac{\log 2}{\tau} \cdot P_i $$ $$ \frac{d s}{dt} = \alpha \cdot \sum \limits_{i=1}^n{P_i} - \beta \cdot s $$

The steady state solution is then:

$$ s_{ss} = R^{-1}(\frac{\log 2}{\tau \cdot r_i})$$ $$ \sum \limits_{i=1}^n{P_{i,ss}} = (\frac{ \beta \cdot s_{ss}}{\alpha})$$
In a simplified way, if we had two identical plasmids: $$ P_{1,ss} + P_{2,ss} = K $$

It is noticeable that plasmid levels are not independent and has a determined value: they depend on each other. However, the value for a system is not known directly using this formula. A hypothesis to explain this solution is that small perturbations can increase the relative level of one plasmid, favouring its growth. As they are repressed as a function of the total number of plasmids but they growth depending on their number, the system is prone to enhance one over another. $$ \frac{d P_1}{d P_2} = \frac{r_1 \cdot P_1}{r_2 \cdot P_2} $$
If both growth rate are constant, both plasmids are going to growth only depending of its level, then, the slope could be calculated using the initial conditions:
$$ m = \frac{P_{01}}{P_{02}} $$


Quasi-steady state approach
To simplify this set of equations, inhibitor dynamics are thought to be faster than plasmid growth, so RNA levels are a direct representation of plasmid levels.

$$ \frac{d P_i}{dt} = r_i \cdot P_i \cdot R(P_i) - \frac{\log 2}{\tau} \cdot P_i $$

Dependence on initial conditions:
To test the previous hypothesis, we integrated the set of two equations using numerical methods as we changed the initial number of each plasmid. As expected, small differences in proportions were amplified.

Stochastic model

The previous model is suitable when the system is not noisy. However, this is not the case and small variations in plasmid levels could be highly amplified. Hence, we modeled the effect of noise in plasmid growth and how it affects. In order to obtain we used the Stochastic Simulation Algorithm (SSA) developed by Gillespie (Erbran et al, 2007 is a good paper about the topic). In order not to enter into details and giving a general overview of the process, SSA simulates by iterations which processes of a given set of reactions occur at which time using a random number generator.

First of all, we developed a code using this algorithm, considering the growth of two plasmids and cell division as independent process. The steps are the following:

Calculation of probabilities: $$ \alpha_{P_i \rightarrow P_i+1} = \frac{1}{r_i \cdot P_i \cdot R(P_i)}$$

Calculation of the timestep when the next reaction occurs: $$ t = \sum \limits_{i=1}^{n}{\alpha_{P_i \rightarrow P_i+1}} \cdot \log(\frac{1}{\theta_1})$$ where $\theta_1$ is a random number from a uniform distribution U(0,1).

Calculation the next reaction. The next reaction taking place will be the one which: $$ \sum \limits_{i=1}^{j}{\alpha_{P_i \rightarrow P_i+1}} < \theta_2 \leq \sum \limits_{i=1}^{j+1}{\alpha_{P_i \rightarrow P_i+1}} $$

Plasmid partition after cell division at t = $\tau$

Here is a couple of examples of what can be obtained for two different conditions: the first one takes places when $P_1=P_2=10$ and the second one when $P_1 = 10$ and $P_2 = 50$

As expected, the deterministic model gives a mean version of the dynamics. However these are not the single cell dynamics, which show oscillatory behaviour. Although the levels of each plasmid can vary, their sum remains unchaged (black line)


A lineage model
Aiming to obtain a better description of our system and perform a more exhaustive analysis, we developed a modeling approach to study the plasmids level in a lineage from a single cell.
As tracking bacteria and their plasmid levels is an important issue, we implemented a tree plot in our code using a matlab package
Partition of plasmids...

Also we are able to track the plasmids levels of each bacteria at the time of their division.

The distribution will be the following for the begining and for the ending of the simulation:

Analysis of plasmids levels...


A step beyond: Resistance

Also we are able to track the plasmids levels of each bacteria at the time of their division.

Approaches for less computational expensive
Although these simulations are powerful and helpful and yield informative results, they are computationally expensive. This could become an important pitfall when dealing with a huge number of cells or long times. We could bypass the problem using an approach if we are working with populations as explained in (Paulsson, 2001). In order to do so, we added a degradation process for each plasmids following this equation:
$$ \alpha_{P_i \rightarrow P_i-1} = \frac{log(2)}{\tau} \cdot P_i$$ In this new system we are observing the global dynamics for a cell population. If there is no resistance, we could obtain the following dynamics, where a plasmid is lost:

This new approach could lead to study cell dynamics in a population.


Conclusions

In this work we aimed to bring the attention to a relevant issue in synthetic biology: plasmid dynamics. Yet they are widely used, we lack of great knowledge about the interaction between two plasmids in the same cell. Plasmids are used because they can perform the tasks we want. However, are their behaviour is the expected one? A relevant example is Marguet et al., 2010 where they obtained an emergent behaviour as result of an unexpected relation between the bacterial physiology and the plasmid dynamics.
In first place, we look for studies with two plasmids in a cell but we were unable to find, although there are interesting work in single plasmid dynamics. Using these models we switch between deterministic and stochastic models to learn about the unspecific interaction between two plasmids in a cell. Therefore, we have developed a new framework to work on this topic. Moreover, we have studied this trying to shed light on this issue.
As we could have learnt, we found that this system is unstable and the only way to keep the plasmid levels is introducing a resistance. However, it is not a good way to maintain those levels: bacteria which have non-desired levels of plasmids are killed. This leads to a lower yield. Also, there is an important variability, which can drive the system away from the expected behaviour. Hence, new...
Conclusions and relevance of this work: New insights in orthogonal plasmids.
Prospective work and further solutions. Modelling for reliable engineered biological systems. Statistical mechanics for plasmids levels.




--REFERENCES----

Erban R, Chapman SJ and Maini PK (2007) A PRACTICAL GUIDE TO STOCHASTIC SIMULATIONS OF REACTION-DIFFUSION PROCESSES link

Klumpp E (2011) Growth-Rate Dependence Reveals Design Principles of Plasmid Copy Number Control PLoS One Vol 6 Issue 5 link

Marguet P, Tanouchi Y, Splitz E, Smith C and You L (2010) Oscillations by Minimal Bacterial Suicide Circuits Reveal Hidden Facets of Host-Circuit Phisology PLoS One Vol 5 Issue 7 link

Paulsson J and Ehrenberg M (2001) Noise in a minimal regulatory network: plasmid copy number control Cambridge Press link