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

From 2012.igem.org

(Difference between revisions)
Line 12: Line 12:
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
</html>
</html>
 +
 +
<h1 id="Associative Memory">Associative Memory</h1>
 +
 +
<h1 id="objective">Introduction</h1>
 +
<p>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 with a high rate.</p>
 +
 +
<h1 id="model">Mathematical Model</h1>
 +
<p>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:</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} \\
+
\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}
 +
 +
<p> 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, table 1. For example, the values for $K$ and $r$ were determined by
 +
examination of the growth curve. </p>
 +
 +
{{: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 }}
 +
 +
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 *********ref***********. We assumed that type A up-regulated cells becomes down-regulated by B with the rate
 +
$\phi_B$ and vice-versa.
 +
\begin{align}
\begin{align}
Line 31: Line 63:
\end{align}
\end{align}
 +
<h1 id="model">Results</h1>
 +
We first evaluated the fraction of cells up-regulated as a function of carrying capacity and of the ratio $\frac{\phi_A}{\phi_B}$,
 +
 +
{{: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 }}
 +
 +
The carrying capacity is an important parameter of the model. For values up to $10^8$, in the equilibrium, no population can reach
 +
quorum since the density of cells is too low, figure *****xis*****. On the other hand, for values higher than $10^8$ the population
 +
that has a higher repressive rate, represented by $\phi$, reachs quorum and represses the other population.
 +
 +
<h1 id="model">Equilibrium points </h1>
 +
 +
At the equilibrium point we have:
\begin{align}  
\begin{align}  

Revision as of 19:57, 26 September 2012

Retrieved from "http://2012.igem.org/Team:USP-UNESP-Brazil/Associative_Memory/Modeling"