Team:USP-UNESP-Brazil/Associative Memory/Modeling

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<h1 id="Associative Memory">Associative Memory</h1>
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<h1 id="objective">Introduction</h1>
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<p>We introduced a mathematical model for two populations of bacteria that interact via quorum sensing. Each population
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produces its own quorum sensing molecule (QSM) and the QSM of one population can be repressive or excitatory to the other
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population in a mechanism analogous to a neuron communication. In our case, a neuron is represented by a population of
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bacteria and a synapse by a communication via QSM. In our analogy, a neuron is activated when the majority of the population
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is in quorum, which means producing the QSM with a high rate.</p>
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<h1 id="model">Mathematical Model</h1>
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<p>Ward et al [1], introduced a mathematical model to describe the growth of populations of bacteria consisting in cell
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that can be either up-regulated or down-regulated. An up-regulated cell  produces QSM faster than a down-regulated cell which
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produces it in a basal rate. If the most bacteria in the population is up-regulated we say the population is in quorum.
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The model consists in three differential equations:</p>
\begin{align}
\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_{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_{1d} - \beta N_{u} \\
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\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 \\
\frac{d}{dt} A &= \kappa_{u} N_{u} + \kappa_{d} N_{d} - \alpha A N_{d} - \lambda A \\
N_{t} &= N_{d} + N_{u}
N_{t} &= N_{d} + N_{u}
\end{align}
\end{align}
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<p> where $N_d$ and $N_u$ are the down-regulated and up-regulated subpopulations density (number of cells per volume),
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$A$ is the density of QSM, $\kappa_{d}$ and $\kappa_{u}$ are the QSM prodution rate of down-regulated and up-regulated, respectively.
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The variable $\lambda$ is the degradation rate of the QSM and $r$ is the cell-division rate.
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It is assumed that down-regulated cells are up-regulated by QSMs with rate constant $\alpha$ and up-regulated becomes down-regulated
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with the rate $\beta$.
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The authors designed some experiments in order to estimate the constants, table 1. For example, the values for $K$ and $r$ were determined by
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examination of the growth curve. </p>
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{{:Team:USP-UNESP-Brazil/Templates/RImage | image=Circularization.jpeg | caption=Fig. 1. Excision recombination reaction steps for CRE and FLP. All reactions are reversible and the arrows represent the forward and backward reactions. | size=600px }}
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In our model, there are two different types of population of bacteria and each type has his own QSM, represented by $A$ and
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$B$. In order to evaluate an interaction between the two type of bacteria we introduced a term in the model proposed by
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Ward et al *********ref***********. We assumed that type A up-regulated cells becomes down-regulated by B with the rate
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$\phi_B$ and vice-versa.
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\begin{align}
\begin{align}
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\end{align}
\end{align}
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<h1 id="model">Results</h1>
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We first evaluated the fraction of cells up-regulated as a function of carrying capacity and of the ratio $\frac{\phi_A}{\phi_B}$,
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{{:Team:USP-UNESP-Brazil/Templates/RImage | image=Circularization.jpeg | caption=Fig. 1. Excision recombination reaction steps for CRE and FLP. All reactions are reversible and the arrows represent the forward and backward reactions. | size=600px }}
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The carrying capacity is an important parameter of the model. For values up to $10^8$, in the equilibrium, no population can reach
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quorum since the density of cells is too low, figure *****xis*****. On the other hand, for values higher than $10^8$ the population
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that has a higher repressive rate, represented by $\phi$, reachs quorum and represses the other population.
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<h1 id="model">Equilibrium points </h1>
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At the equilibrium point we have:
\begin{align}  
\begin{align}  

Revision as of 19:57, 26 September 2012