Assessment of strategies for individual-based modeling of bacterial conjugation

Antonio Prestes García, Alfonso Rodríguez-Patón


The bacterial conjugation, despite of its importance, is poorly understood systemically. In this work we propose a novel framework for the individual-based modeling which hinge on two main key points, namely the cell-cycle timing when conjugative is most likely to occur and the metabolic penalizations incurred by donor cells during the conjugative transfer and by the transconjugants cells once they have been infected and have to express the required gene products for the plasmid housekeeping. We have evaluated the model predictions using eight different plasmids on E. coli host.


[[TAG] CONTEXT] The Domain Bacteria encompass one of most diverse and abundant form of life on earth. Part of this diversity is certainly in part a direct consequence of a succinct genome complemented by the existence of a feature rich supra-individual gene pool which is readily available for individuals in a population through different mechanisms. One of these mechanisms is the bacterial conjugation which is basically a form of horizontal gene transfer where cluster of genes are transferred from cell to cell in some population. The plasmids, which are the fundamental unit of horizontal gene transfer, are circular double stranded DNA and they are also autonomous replicons replicating independently of bacterial chromosome and having their own life-cycle.

Plasmids can also cross the domain boundaries and infect eukaryotic cells, as can be observed in genus Agrobacterium, responsible for causing plant diseases. That is the case of A. rhizogenes and A. tumefaciens and their associated plasmids which are termed Ri and Ti standing respectively for root inducer and tumor inducer plasmids. These plasmids are responsible for hairy root and crown gall plant diseases respectively. On the other hand the genus Rhizobium and its associated plasmids induce the nodule formation on plant roots acting as symbionts in the atmospheric nitrogen fixation process. These plasmids can also be harnessed for the insertion of T-DNA in plants to create transgenic cultivars. The bacterial gene pool is also used for genetic engineered plant cultivar creation such as herbicide resistant transgenic plants. That is the case of aroA gene coding the AprA enzymes which makes the plant show tolerance to glyphosate. Last but not least important, plasmids are deemed to be the main cause of spreading the multi-drug resistance associated with bacterial populations exposed to the antibiotic selective pressure. In fact this severely limits the arsenal of drugs available to fight against bacterial infections.

[[TAG] NEED] Conjugative plasmids is beginning to seen as a viable tool as the wiring protocol for population wide computations, as the basic bricks for bacterial nano-networks and for more complex applications in synthetic biology but despite of the high relevance, there are either no dependable technique readily available yet, which have been thoroughly tested and systematically validated against the experimental data or simply an accepted standard to model the plasmid spread dynamics using single cell resolution. There is also more open question than answers on many points of the lateral gene transfer process with some opposing views about some specific aspects.

[[TAG] TASK] We are primarily concerned, in this work, with providing a robust operational model for conjugation using an individual-based approach which can be easily adapted and used a standard modeling tool for simulating the kinetics of conjugative plasmids. In order to accomplish that goal we must bring to light some hidden aspects of conjugation which cannot be observed in whole population experimental setups and only can be detected at a single individual resolution. This approach has an added value because at the same time we produce a more dependable model we are providing useful hints about the local intra intracellular behavior of conjugation which certainly is useful to understand the process.

That is not an easy task because we have to make many assumptions and simplifications to provide a usable abstraction for the process. It has been used in other works as the operational abstraction for bacterial conjugation, a set of rules relying on parameters like some arbitrary probability value, the pilus scan speed, the action radius of conjugative pili(Merkey 2011) or even simply the number of infected individuals on the neighborhood(Krone 2007).

As general rule good initial assumptions for individual-based models are those which are biologically consistent and could be almost axiomatically accepted. The assumptions which fall in this category are fundamentally that most of processes inside of any cell are uphill which basically means that they have fight against an energy gradient, in other words they have a cost. The second assumption is that cellular processes are subjected to a precise set of timing constraints for all cellular activities and any deviation on these timers is disruptive for cellular activities.

In order to thoroughly understand how the plasmids are distributed and evolve in a bacterial population is necessary to separately identify the different aspects that affect the progression of cell to cell transmission and the invasion in the whole population. The final outcome of the process leading to the partial or total infection a bacterial colony can be seen as the sum of a set of contributions due to vertical and horizontal transfers as well as how much the metabolic burden contributes, as a negative feedback loop, to the pace of conjugative process.

[[TAG] OBJECT OF THE DOCUMENT] In this work we introduce an individual-based model for bacterial conjugation constructed using a modular design which has been used to evaluate the better alternative for modeling and understand the conjugation systemically. Thereby, using this modular design we have plugged different approaches to model the conjugation with respect the time within cell cycle which produces the best and most natural fit to a complete and diverse experimental data.

Materials and Methods

Theoretical Framework

The wave speed of plasmid spread in a bacterial population is a complex and non-lineal process and as such is hard to grasp some intuitive idea about the main factors controlling the global behavior of the system as a whole. In order to understand the spatiotemporal behavior and visualize the processes some simplifications are required to be made. Hence we have used a network analogy, comparing bacterial cells to network nodes which store and forward messages, being the conjugative plasmids the message transmitted from cell to cell.

It is known the most cellular processes follows a tight time schedule and is natural to think that the bacterial conjugation must be subject to the same constraints. Nonetheless until now, no works have studied how conjugative events are distributed across the cell cycle duration. With respect to time and ignoring other constraints, such as the availability of reachable susceptible individuals, we may state the following biologically relevant hypotheses:

  • The time of conjugative transfer event is completely independent of bacterial cell cycle and grown conditions being merely a function of plasmid endogenous factors.

  • On the other hand, plasmids could behave as all other cellular processes being linked to the host metabolic level and growth rate.

The main implication of first hypothesis, as should became clear in the next paragraphs, is that the larger is the bacterial generation time more efficient becomes the conjugative process because the infection rate surpasses the host growth rate allowing the complete plasmid invasion. In the second hypothesis, the efficiency of a conjugative plasmid depends on the host growth regime and also on the plasmid intrinsic factors. Hence the time where conjugation will occur can be expressed as a percentage of bacterial doubling time indicating the fact that conjugative activity depends and is coordinated by the host dynamics.

It has already been pointed out recently that timing and delays are the most important parameters governing the global observed dynamics of plasmid spread in bacterial populations(Merkey 2011). This work has shown using sensitivity analysis, that the lag time from the initial infection until the new formed transconjugant cell becomes a proficient donor is the most significant parameter of the model.

The bacterial cellular dynamics controlling the growth and division process has a theoretical framework which has been pointed out as the central dogma bacterial cell division cycle or simply of Cooper-Helmstetter model(Zaritsky 2011). The model describes the bacterial cell process relating a set of events such as the chromosome replication and division to the changes in the cell mass.

Therefore, taking into account the bacterial cell cycle, three basic cases for the temporal properties of bacterial conjugation which are enumerated bellow:

  • Conjugation is independent of host dynamics and conjugation may take place as soon a suitable recipient is found on the neighborhood of plasmid bearing cell.

  • The frequency of conjugative events is evenly distributed across the cell cycle.

  • Finally, conjugation is linked and coordinated to the host dynamics being the frequency of conjugative events concentrated in some specific point late in the cell cycle.

The time related parameters of a conjugation model can be grouped, for the sake of simplicity, as a plasmid infection forwarding delay which stands for the lag time required for some plasmid being forewarned to any adjacent cell. Thereby we can study the forwarding delay effect on the slope of flooding speed.

In the most elemental level, the rate at which the infection progresses depends on how many times every single cell can spread the plasmid, which hereafter we call intrinsic conjugation rate, it also depends on how fast the transmissions from donor cells to recipient cells and further retransmission from transconjugants cells can be accomplished. Thus the intrinsic conjugation rate, ignoring other aspects, depends almost exclusively on the bacterial growth regime and on the time required for a complete plasmid transfer. Just to put forward a simple example to clarify that idea: Using a generation time \(\mathcal{G}=20^\prime\), the average transfer speed, which is approximately 45kb/m and the archetypical F plasmid which has a size of 100 kb, the minimal amount of time required for conjugation to complete would be roughly two minutes and the maximum intrinsic transfer rate would be 10 conjugations per cell cycle.

Let us assume a simple and yet idealized bacterial population of \(N\) individuals with a growth rate \(\mu = 0\). The population will be sessile, which implies that the topology of network will be static and individuals cells will be distributed side by side conforming an interaction structure which could be abstracted by a linear graph as can be seen in Fig.  \ref{fig:conjugation-delays}.

Hence assuming the functional similarity between conjugation and networks a common metrics can be used to analyze the process. One of these metrics is the total delay required for flooding all nodes in the network with some message \(\mathcal{M}\) which in the bacterial context is given by the Equation  \eqref{eq:delay}

\[\label{eq:delay} \tau = N \times T + (H - 1) \times P + (H - 1) \times T\]  

being N the number of messages being forwarded, T the time required for transmitting the message, P the time needed for each node to process the message before being able to forward it again and finally H stands for the number of hops in or network. Putting it in terms of bacterial conjugation we have that N represents the plasmid size in kilobases, T the time required to move a single kilobase from a cell to another, P may represent the two sources of delay which are referred as recovery and maturation time for donors and recipient cells respectively. Finally H is the actual number of bacterial cells in the linear topology already mentioned.

The aggregated value referred globally as maturation time includes, amongst other factors, (a) the time required for adding the second strand to the plasmid inside the new formed transconjugant cells, (b) the time required to express all genes coding the proteins for the trans-envelope apparatus, and (c) possibly the effect of SOS response triggered by entrance of single stranded DNA into the transconjugant cell(Baharoglu 2010).

The Figure  \ref{fig:interaction-graph} shows the temporal evolution of the linear graph which denotes the bacterial population being infected. In this graph every donor cell, depicted by \(B^t_i\) only have a reachable neighbor which can be infected in a conjugative event. In other words the plasmid bearing cell \(B^t_i\) can only interact and conjugate with the bacterial agent \(B^r_{i+1}\). Thus, in the simplest case where the forwarding delay is fixed, the infection speed can be approximated by a straight line in a form of \(y = \alpha + \beta x\) being \(\alpha\) the intercept and \(\beta\) the slope. The value of slope for the most general case is given by the expression shown in Equation  \eqref{eq:slope}