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\section{Abstract}  The bacterial conjugation, despite of its importance, is poorly understood systemically. In this work we propose a new 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.  \section{Introduction}  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 {\it Agrobacterium}, responsible for causing plant diseases. That is the case of {\it A. rhizogenes} and {\it 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.  Conjugative plasmids are also the basic bricks 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 just an accepted standard to model the plasmid spread dynamics using single cell resolution.   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 model abstraction for conjugation, a set of rules relying on parameters like some arbitrary probability value, the pilus scan speed, the action radius of conjugative pili\cite{citeulike:10283930} or even simply the number of infected individuals on the neighborhood\cite{citeulike:3567840}.     As general rule good initial assumptions 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 for healthy cells are subjected to some precise set of timing constraints for all cellular activities and any deviation on timers is disruptive for cellular activities.   In this work we introduce an individual-based model of bacterial conjugation constructed using a modular design which has been used to evaluate the better alternative for modeling and understand the conjugation systemically.  \section{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 processes some simplifications are required to be made. 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:forwarding-delay}.   \begin{figure}  \centering  \includegraphics[scale=0.6]{figures/paper-20(LinearGraph-0)v2.pdf}  \caption[Forwarding delay]{\label{fig:forwarding-delay} {\bf The forwarding delay scheme}. The $B^r$ and $B^t$ are respectively the recipient and transconjugant bacterial nodes. The figure shows the meaning of forwarding delay which is the time elapsed since a bacteria $B^r_i$ is infected becoming a transconjugant $B^t_i$ and infect the next recipient cell $B^r_{i+1}$ in the linear graph shown.}  \end{figure}  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}  \  \begin{equation}  \label{eq:slope}  \beta = \frac{|B^t|t_i - |B^t|t_{i-1}}{\Delta_t}, \forall i > 0  \end{equation}  \  where $|B^t|t_i$ and $|B^t|t_{i-1}$ are respectively the counting of $B^t$ cells at $t_i$ and $t_{i-1}$ and $\Delta_t = t_i - t_{i-1}$. The The intercept $\alpha$ is zero since at $t_0$ the number of $|B^t|$ cells are also zero. The choice of $|B^t|$ as the indicator of network flooding is just for the sake of generality but the same principles will also hold if instead of it the value of $T/(T+R)$ were used.  \begin{figure*}  \centering  \includegraphics[scale=0.61]{figures/paper-20(LinearGraph-1)v2.pdf}  \caption[Interaction graph]{\label{fig:interaction-graph} {\bf The simplified interaction graph}. This figure shows from left to right the snapshots for the temporal evolution of the linear graph representing our bacterial population. The cell types are $B^r$, $B^d$ and $B^t$ standing respectively for recipients, donors and transconjugants.}   \end{figure*}  The Equation ~\eqref{eq:slope}, in the case of unitary increments in the infected nodes which is exactly what is happening in the linear graph show in Figure ~\ref{fig:interaction-graph} , can be simplified to $\beta = 1/\Delta_t$. Withal it is easy to realize that the slope have the following $\lim_{\Delta_t \to \infty} 1/\Delta_t = 0$ $\lim_{\Delta_t \to 0} 1/\Delta_t = \infty$ lower and upper limits.   \begin{figure*}  \centering  \includegraphics[scale=0.8]{figures/forwarding-delay.pdf}  \caption[Forwarding delay effect]{\label{fig:forwarding-delay} {\bf The effect of forwarding delay in infection speed.} This graphic shows the effect of increasing delays in the plasmid forwarding network. As can be observed the slope is inversely proportional to the forwarding delay. }   \end{figure*}