Discussion
Replication
Very few is addressed with respect to replication other than consider the plasmid copy number, modelers then to ignore the replication process and assume that it is instantaneous, and do not even care mention.
This could be a consequence of that in general scientist study populations level effects of plasmids processes. But we can expect as single-cell perspective becomes more popular, plasmid replication would be described more carefully.
Prove of the relevance of this process is shown in \citep{Kentzoglanakis_2013} were they consider plasmid replication timing to be an important issue on the generation of segregants and thus, in the stability of plasmids.
Segregation
The process of plasmid segregation, in general, has been underestimated. In many studies it is intentionally not considered. Although, in the great majority of models revisted (regarding non-conjugative plasmids), segregation is taken into account in the rate of generation of plasmid free cells and is either explored or parameterized from experimental data, plasmids are considered to be segregated randomly to the daughter cells. And it is proven that it is not the case for high copy number plasmids \citep{M_nch_2019}.
An great example of this can be seen in \citep{Ayala_Sanmartin_1989} were they related plasmid copy number to a degree multimerization and analyze stability of multicopy plasmids using a model that failed to predict stability for high copy plasmids that presented dimerization. This could be explained because the model is assuming random segregation of plasmids, which may not be the case for plasmids presenting dimerization. Insights of this hypothesis are supported by \citep{Reyes_Lamothe_2013} where they show by using a model and microscopy observation that high copy plasmids are mainly found in the poles due to a displacement by the nucleoid. \citep{Hsu_2019} they use the FROS technique and a probabilistic model based on plasmid localization to state that plasmids clusters in poles and some can move along the cell to form another cluster in the other pole and this behavior can explain heterogeneity of segregation. And in \citealp{M_nch_2019} they state that high copy plasmids present unequal segregation.
Conjugation
The findings by \cite{Levin_1979}, states that conjugation rate should be addressed by a function dependent on substrate concentration, the authors in \cite{Simonsen_1990} consider conjugation rate to be constant under batch experiments , regardless that they based their model in the former.
In this matter, other studies that also implement conjugation as constant rate, find important to make adjustments to their models to fit lag and stationary phase conjugation rates \cite{Philipsen_2010,Yurtsev_2013}. And in the case of \citep{Merkey_2011} found that a growth(resource) dependent conjugation rate is of vital importance in successfully model spatially structured environments.
There is another subtle discrepancy concerning interactions between conjugative and non-conjugative plasmids. In the work of \citep{Levin_1980} they conclude that maintenance of non-conjugative plasmids requires selection even when there is a mobilizing plasmid. In counterpart, \cite{Werisch_2017} states the opposite using a different approach in a almost the same case of study, the only difference is that in the latter the plasmids are incompatible. It is remarkable how this "little" difference can undergo really opposite conclusions.
Compensation
The mechanisms of plasmid cost amelioration could be due to changes in genes in expression of either in chromosomal or plasmid genes, changes in conjugation rates or the lost of plasmid genes \citep{Zwanzig_2019}. In this same work they explore the influence of compensatory mutations in the stability of plasmid, proving a fact that was often assumed, that those carried in plasmids has a greater effect due to horizontal propagation. But the relevance of considering compensatory mutations in modelling plasmid stability is very well exemplified in \citep{Millan_2014}, where the model failed to describe the stability unless compensatory mutation are addressed.
Compatibility
The most used scheme in modelling plasmid compatibility is surface exclusion through a state variable \citep{Gregory_2008}, although some authors prefer to appeal to replication machinery competition to model uniqueness of a plasmid type within a cell or population \citep{Summers_1996}.
The model developed in \citep{Gregory_2008}, has been proven to be a good approach for study conjugation and compatibility of plasmids on spatially structured environments, and thus could be used for exploring more realistic situations, or extended for studying communities, for example.
An interesting remark about plasmid compatibility is declared in \citep{Werisch_2017}, throughout simulations they find that under spatially structured environments, non-transmissible plasmids can be maintained in the population if a non-compatible conjugative plasmid is present, even when the non-transmissible plasmid provide no advantage to the host, and this could not be achieved if the conjugative plasmid is compatible. This finding could be used to explain the existence of large non-transmissible plasmids, giving some light to the plasmid paradox.
Stability
The majority of plasmid related model embrace plasmid stability, although many of them postulate a simple equations for the fraction of plasmid bearer population in very simplistic manner. More complex situations has been addressed.
In general, the studies regarding plasmid stability states that there is a trade-off between the segregation rate and the plasmid cost, strategies could be implemented such as selection against the plasmids, the use conjugative plasmids \citep{Werisch_2017}.
These strategies are relevant in the elucidation of the plasmid paradox which is major concert interest for many biologist. In the light of bioengineering, this strategies entails rather costs of production or production inefficiency. For this reason, finding easy implemented schemes that promotes plasmid stability are particularly relevant. The findings of \citep{Yuan_2010} propose a feasible strategy for the maintenance of non conjugative plasmids: controlling the substrate concentration by periodic pulses , dilution rate.
Although the work presented in \cite{Yurtsev_2013} disregarded the formation of plasmid free cells. It gave us a philosophical contribution to keep in mind when attempting to artificially stabilize a plasmid using inhibitors: the social aspect of the antibiotics and the importance of characterizing the social interactions such as cross-protection or any other public good.
The general three factors contributing to the maintenance of plasmids are horizontal gene transfer, selection against the trait carried by the a plasmid and compensatory mutations \citep{Millan_2014}.
PSK
PSK systems are per se mechanisms to ensure plasmid maintenance, and thus plasmid stability. Although this strategy is very effective, it is not often used in bioengineering because it is costly and reduces product efficiency.
We have analyzed \citep{Willms_2006}a model that explored the effects of the efficiency of the PSK with rather logic results, the efficiency increases plasmid persistence. But future applications of this model can be still explored. More over, we believe that there is still work to do in this matter, for example, study PSK in high copy number plasmids, particularly analyzing the dynamics of plasmid and toxin segregation at single-cell level. Or their affect on communities when the toxin is and exported bacteriocin.
Modelling approaches
Ordinary differential equation models has been largely proven to satisfactorily describe population dynamics. Then popularity of this models not only rely on the simplicity or the diversity of methods to find analytical solutions, and thus, capture the whole picture of the dynamics of a system. They are also relative simple to develop and to compute numerical solutions.
But their greater advantage is that they are based on the mass action law which allow the modelers to simply processes and the equations. For this reason this type of models are fit to explain experiments under laboratory conditions specially in chemostats where everything is "well-mixed". But it could not be the case nature occurring population where we often find spatial structure, for modelling this scenarios partial differential equation should be accounted and this are difficult to solve.
We have seen in \citep{Ponciano_2006} that stochastic models describe plasmid dynamics better than deterministic models (which are mechanistically driven) from strategies incorporating noise directly from observed data. This gives the expectation that this kind of models are better fitted to study plasmids when not much of their biology is known.
As experimental techniques becomes more sophisticated, such as the using of reporter genes and imaging techniques, more in situ specific data can be taken into account for the study of plasmid dynamics in a more realistic way. The general assumption of mass action of population and stochastic models only works for well mixed environments. Which is not the case on spatially structure nature systems. In this scenarios, ecological interactions between populations could be better understood using individual-based models. \citep{S_rensen_2005}
This modelling framework can integrate individual specific characteristics and as the the number of processes involved increases so does the programming and computational complexity. Nonetheless, this approach becomes very useful in studying communities ecological interactions in spatially structured manner. The advantage of this models is that they can efficiently capture population dynamics from individuals interactions, that at the same time this interactions are driven by properties that could or not vary between individuals of the same type. Thus facilitating the incorporation of biological noise such as different cell cycle states or metabolic variations.
Advances in technology and experimental approaches has increased the level comprehension of the molecular mechanisms involved in plasmid dynamics modelling. An example of this is how the grade of complexity in which replication and segregation mechanisms has been addressed over time. On the other hand more refined models will help in elucidating the behavior of plasmids dynamics. This impose a feedback loop between models and experimental knowledge.
Another technological relevant factor to consider is the computational power, some modelers often decide to ignore a particular process with the assumption that it does not change qualitatively their results. This practice definitely simplify models, which make them easier to resolve analytically or to perform simulations. We believe that, at least computational problems, should not be factor to be considered anymore. But of course, simplifications and generalizations, would be addressed depending on the objectives of the research investigation and the data or experimental limitations.
We will inquire that the relevance of models is that they can be used to make predictions of plasmids behavior when we have biological data limitations.