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Introduction

Project Overview

Plasmid
Plug&Play

Associative Memory Network

Extras

Introduction

Project Overview

Plasmid Plug&Play

Associative Memory
Network

Extras

Associative Memory

We introduced a mathematical model for two populations of bacteria that interact via quorum sensing. Each population produces its own quorum sensing molecule (QSM) and the QSM of one population can be repressive or excitatory to the other population in a mechanism analogous to a neuron communication. In our case, a neuron is represented by a population of bacteria and a synapse by a communication via QSM. In our analogy, a neuron is activated when the majority of the population is in quorum, which means producing the QSM at a high rate.

Mathematical Model

Ward et al [1] introduced a mathematical model to describe the growth of populations of bacteria consisting in cell that can be either up-regulated or down-regulated. An up-regulated cell produces QSM faster than a down-regulated cell which produces it in a basal rate. If the most bacteria in the population is up-regulated we say the population is in quorum. The model consists in three differential equations:

\begin{align} \frac{d}{dt} N_{d} &= rN_{t}\Big[1 - \frac{N_{t}}{K}\Big] - \alpha A N_{d} + \beta N_{u} \\ \frac{d}{dt} N_{u} &= \alpha A N_{d} - \beta N_{u} \\ \frac{d}{dt} A &= \kappa_{u} N_{u} + \kappa_{d} N_{d} - \alpha A N_{d} - \lambda A \\ N_{t} &= N_{d} + N_{u} \end{align}

where $N_d$ and $N_u$ are the down-regulated and up-regulated subpopulations density (number of cells per volume), $A$ is the density of QSM, $\kappa_{d}$ and $\kappa_{u}$ are the QSM prodution rate of down-regulated and up-regulated, respectively. The variable $\lambda$ is the degradation rate of the QSM and $r$ is the cell-division rate. It is assumed that down-regulated cells are up-regulated by QSMs with rate constant $\alpha$ and up-regulated becomes down-regulated with the rate $\beta$. The authors designed some experiments in order to estimate the constants, Figure 1. For example, the values for $K$ and $r$ were determined by examination of the growth curve.

Fig. 1. Parameter values obtained by Ward et al.

In our model, there are two different types of population of bacteria and each type has his own QSM, represented by $A$ and $B$. In order to evaluate an interaction between the two type of bacteria we introduced a term in the model proposed by Ward et al [1]. We assumed that type A up-regulated cells becomes down-regulated by B with the rate $\phi_B$ and vice-versa.

\begin{align} \frac{d}{dt}N_{Ad} &= rN_{At}\Big[1 - \frac{N_{At}}{K}\Big] - \alpha_A A N_{Ad} + (\beta_A + \phi_B B)N_{Au} \\ \frac{d}{dt}N_{Bd} &= rN_{Bt}\Big[1 - \frac{N_{Bt}}{K}\Big] - \alpha_B A N_{Bd} + (\beta_B + \phi_A A)N_{Bu} \\ \frac{d}{dt}N_{Au} &= \alpha_A A N_{Ad} - \beta_A N_{Au} - \phi_B B N_{Au}\\ \frac{d}{dt}N_{Bu} &= \alpha_B B N_{Bd} - \beta_B N_{Bu} - \phi_A A N_{Bu} \\ \frac{d}{dt}A &= \kappa_{Au} N_{Au} + \kappa_{Ad} N_{Ad} - \alpha_A A N_{Ad} - \lambda_A A - \phi_A A N_{Bu} \\ \frac{d}{dt}B &= \kappa_{Bu} N_{Bu} + \kappa_{Bd} N_{Bd} - \alpha_B B N_{Bd} - \lambda_B B - \phi_B B N_{Au} \\ N_{At} &= N_{Ad}+N_{Au} \\ N_{Bt} &= N_{Bd}+N_{Bu} \end{align}

Results

We first evaluated the fraction of up-regulated cells as a function of the carrying capacity ($K$) and of the $\frac{\phi_A}{\phi_B}$ ratio, and $K$ turned out to be an important parameter of the model. For values of $K$ up to $10^8$, in the equilibrium, no population can reach the quorum state, since the density of cells is too low. On the other hand, for values higher than $10^8$ the population that has a higher repression rate, represented by $\phi$, reaches quorum and represses the other population, as presented in Figure 2.

Fig. 2. The fraction of the up-regulated population as a function of the carrying capacity ($K$) and the ratio $\frac{\phi_A}{\phi_B}$ for the case $\phi_B = \alpha$, at equilibrium. Initial conditions: $N_{Au} = N_{Bu} = A = B = 0$.

Equilibrium points

At equilibrium we have:

\begin{align} &-\alpha_A A N_{Ad} + \beta_A N_{Au} + \phi_B B N_{Au} &= 0 \\ &\kappa_{Au} N_{Au} + \kappa_{Ad} N_{Ad} - \alpha_A A N_{Ad} - \lambda_A A - \phi_A A N_{Bu} &= 0 \\ &-\alpha_B B N_{Bd} + \beta_B N_{Bu} + \phi_A A N_{Bu} &= 0 \\ &\kappa_{Bu} N_{Bu} + \kappa_{Bd} N_{Bd} - \alpha_B B N_{Bd} - \lambda_B B - \phi_B B N_{Au} &= 0 \\ &N_{Au} + N_{Ad} &= K \\ &N_{Bu} + N_{Bd} &= K \end{align}

These equations can be reduced to two expressions involving $x = N_{Au}/K$ and $y = N_{Bu}/K$:

\begin{align} \hspace{-0.5 cm} \frac{y(\kappa_{Bu} - \kappa_{Bd}) + \kappa_{Bd}}{\alpha_B (1-y) + \lambda^*_B + y\phi_B} = \frac{\Big[x(\kappa_{Au} - \kappa_{Ad}) + \kappa_{Ad}\Big]\alpha_A (1-x)} {\Big[\alpha_A (1-x) + \lambda^*_A + y\phi_A\Big]x\phi_B} - \frac{\beta_A}{\phi_B} \\ \hspace{-0.5 cm} \frac{x(\kappa_{Au} - \kappa_{Ad}) + \kappa_{Ad}}{\alpha_A (1-x) + \lambda^*_A + x\phi_A} = \frac{\Big[y(\kappa_{Bu} - \kappa_{Bd}) + \kappa_{Bd}\Big]\alpha_B (1-y)} {\Big[\alpha_B (1-y) + \lambda^*_B + x\phi_B\Big]y\phi_A} - \frac{\beta_B}{\phi_A} \end{align}

where $\lambda^* = \lambda/K$, and $x$ and $y$ range from 0 up to 1. These expressions are similar, since exchanging $A$ and $B$ turns one equation into the other.

Fig. 3. Each curve represents the solution of one equation, their intersections being the equilibrium points. One is close to $(0,0)$, the other to $(1,1)$.

Thus, the equilibrium points $(x,y)$ are placed in the intersection between the solutions for the relations above. Depending on the set of parameters, one can find two to four equilibria - the first one close to $(0,0)$, representing the repression of both populations, and the second one close to $(1,1)$, representing the activation of both populations, as presented below:

Fig. 4. When $\frac{\phi_A}{\phi_B} \gg 1$, besides the equilibria close to $(0,0)$ and $(1,1)$, there is also a point close to to $(1,0)$.

A third equilibrium point emerges when $\frac{\phi_A}{\phi_B} \gg 1$ - which means the repression of population A over population B is much greater than its counterpart. In this case, the system reaches an equilibrium close to $(1,0)$: population A activated, population B repressed. The behavior is analogous if $\frac{\phi_A}{\phi_B} \ll 1$.

Fig. 4. When $\frac{\phi_A}{\alpha_A} \gg 1$ and $\frac{\phi_B}{\alpha_B} \gg 1$, there are four equilibria, close to $(0,0)$, $(1,1)$, $(1,0)$ and $(0,1)$.}

Finally, when both $\phi_A$ and $\phi_B$ are big when compared to $\alpha_A$ and $\alpha_B$, both populations A and B are able to repress each other, depending on the initial conditions: there are equilibria both close to $(1,0)$ and to $(0,1)$ - with repression of one population and activation of the other.