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

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<h1 id="Associative Memory">Associative Memory</h1>
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<h1 id="model">Mathematical Model</h1>
<p>We introduced a mathematical model for two populations of bacteria that interact via quorum sensing. Each population  
<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  
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  
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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bacteria whereas 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.</p>
is in quorum, which means producing the QSM at a high rate.</p>
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<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  
<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
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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that has a higher repression rate, represented by $\phi$, reaches quorum and represses the other population, as presented in Figure 1.
that has a higher repression rate, represented by $\phi$, reaches quorum and represses the other population, as presented in Figure 1.
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{{:Team:USP-UNESP-Brazil/Templates/RImage | image=KxRphis_alpha.jpg | caption=Fig. 1. 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: $N_{Au} = N_{Bu} = A = B = 0$. | size=620px}}
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{{:Team:USP-UNESP-Brazil/Templates/RImage | image=KxRphis_alpha.jpg | caption=Fig. 1. The fraction of the up-regulated cells, for population A and B, 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: $N_{Au} = N_{Bu} = A = B = 0$. | size=620px}}
<h2 id="model">Equilibrium points </h2>
<h2 id="model">Equilibrium points </h2>
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At equilibrium point we have:
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At equilibrium point, the equations 5 to 10 are equal to zero and thus:
\begin{align}  
\begin{align}  
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{{:Team:USP-UNESP-Brazil/Templates/RImage | image=Phisiguais1.jpg | caption=Fig. 2. Each curve represents the solution of one equation, their intersections being the equilibrium points. One is close to $(0,0)$, the other to $(1,1)$. | size=350px}}
{{:Team:USP-UNESP-Brazil/Templates/RImage | image=Phisiguais1.jpg | caption=Fig. 2. Each curve represents the solution of one equation, their intersections being the equilibrium points. One is close to $(0,0)$, the other to $(1,1)$. | size=350px}}
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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 in Fig. 2. In this case we used the parameters presented in Table 1, for 20% serum solution growth medium. The same behavior was found using the parameters for LB growth medium.  
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 in Fig. 2. In this case we used the parameters presented in Table 1, for 20% serum solution growth medium. The same behavior was found using the parameters for LB growth medium.  
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We limitated the range of the variables $x$ and $y$ to [0,1] since it is the range that has a biological meaning. The entire curve can be seen in the following link [https://2012.igem.org/File:Phisiguais_2jpg.jpeg].
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{{:Team:USP-UNESP-Brazil/Templates/LImage | image=Phis100.jpg | left | caption=Fig. 3. 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 $(1,0)$. | size=350px}}
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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, Fig. 3. In this case, the system also can reach an equilibrium close to $(1,0)$: population A activated, population B repressed. The behavior is analogous if $\frac{\phi_A}{\phi_B} \ll 1$.
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{{:Team:USP-UNESP-Brazil/Templates/RImage | image=Phis100.jpg | caption=Fig. 3. 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 $(1,0)$. | size=350px}}
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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, Fig. 3. In this case, the system reaches an equilibrium close to $(1,0)$: population A activated, population B repressed. The behavior is analogous if $\frac{\phi_A}{\phi_B} \ll 1$.
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{{:Team:USP-UNESP-Brazil/Templates/RImage | image=Phis100ab.jpg | caption=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 $(0,1)$.} | size=350px}}
{{:Team:USP-UNESP-Brazil/Templates/RImage | image=Phis100ab.jpg | caption=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 $(0,1)$.} | size=350px}}
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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
 
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repress each other, depending on the initial conditions: there are equilibria both close to $(1,0)$ and to $(0,1)$ - with repression
 
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of one population and activation of the other, Fig. 4.
 
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Finally, when both $\phi_A$ and $\phi_B$ are large when compared to $\alpha_A$ and $\alpha_B$, both populations A and B are able to
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repress each other. In this case, the system can reach both equilibruim points $(1,0)$ and $(0,1)$ - repression
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of one population and activation of the other, Fig. 4. This is the condition we should find experimentally in order to make our system works properly.
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Its desired that, without any stimulus, the system converges to the point (0,0) and stays at this point. However, with a suficient  stimulus the system should be able to converge to the point (0,1) or (1,0), depending on which population was stimulated.
<h1 id="discussion">Discussion</h1>
<h1 id="discussion">Discussion</h1>
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The Associative Memory Network project was based on a mathematical formulation of a neural network developed in 1982 by John Hopfield [3]. In order to connect the bacteria behavior during quorum sensing to a Hopfield network, we introduced an interaction between two populations in a mathematical model for quorum sensing, in order to represent neuronal repression/activation. Through steady-state analysis, it was possible to find up to four equilibrium points, representing the activation of both populations, activation of one population and repression of the other, and repression of both populations. The existence of these steady-state solutions depends on the set of parameters, and stability analysis is being conducted to answer which regions in parameter space guarantee stability of each equilibria.
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The Associative Memory Network project was based on a mathematical formulation of a neural network developed in 1982 by John Hopfield [3]. In order to connect the bacteria behavior during quorum sensing to a Hopfield network, we introduced an interaction between two populations in a mathematical model for quorum sensing. This interaction represents the neuronal repression/activation. Through steady-state analysis, it was possible to find up to four equilibrium points, representing the activation of both populations, activation of one population and repression of the other, and repression of both populations. The existence of these steady-state solutions depends on the set of parameters and stability analysis is being conducted to answer which regions in parameter space guarantee stability of each equilibria.
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However, numerical simulations indicate that the carrying capacity $K$ is a fundamental parameter in the determination of the stability of quorum-state fixed points: changing its value had no effect on the location of these equilibria in $(x,y)$ space, but Figure 1 shows increasing $K$ leads to quorum. Therefore, even though $K$ does not affect the existence of any fixed points, their stability seems to depend on its value - it would change a steady state from unstable to stable, so that for a population that would not reach quorum at a low carrying capacity might reach it when this parameter is increased.
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However, numerical simulations indicate that the carrying capacity $K$ is a fundamental parameter in the determination of the stability of quorum-state fixed points: changing its value had no effect on the location of these equilibria in $(x,y)$ space but, as shown in figure 1, increasing $K$ leads to quorum as a equilibrium point. Therefore, even though $K$ does not affect the existence of any fixed points, their stability seems to depend on its value - it would change a steady state from unstable to stable, so that for a population that would not reach quorum at a low carrying capacity might reach it when this parameter is increased.
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In summary, the goal has been achieved: it was proven that two bacterial populations are able to interact through repression and activation in order to reproduce a Hopfield Network, and that different combinations of parameters and initial conditions may lead to different activation patterns: both populations activated, both repressed, or the asymetrical case - one up, one down.
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In summary, the goal of this mathematical modeling has been achieved: it was verified that two bacterial populations are able to interact through repression and activation in order to reproduce a Hopfield Network. Additionally, different combinations of parameters and initial conditions may lead to different activation patterns: both populations activated, both repressed, or the asymetrical case - one up, one down.
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Our further steps are to find the conditions for stability, and to investigate patterns on the initial conditions and the stable steady states - in other words, input and output. A network with more than two interacting populations holds a systemic memory capable of storing and responding to a much wider range of patterns - a possibility that is being now explored by our team.
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Our further steps are to find the conditions for stability, which means to investigate the behaviour of the stable steady states as a function of the initial conditions - in other words, how the input changes the output. A network with more than two interacting populations is able to hold a systemic memory capable of storing and responding to a much wider range of patterns - a possibility that can be explored by our team in the future.
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<p>[1] J. P. Ward, J.R. King, A. J. Koerber, P. Williams, J. M. Croft and R. E. Sockett
<p>[1] J. P. Ward, J.R. King, A. J. Koerber, P. Williams, J. M. Croft and R. E. Sockett
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''Mathematical modelling of quorum sensing in bacteria''. </p>
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''Mathematical modelling of quorum sensing in bacteria''.  
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Math Med Biol (2001) 18(3)
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Math Med Biol (2001) 18(3)</p>
<p>[2] http://partsregistry.org/ </p>
<p>[2] http://partsregistry.org/ </p>
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<p>[3] J. J. Hopfield, "Neural networks and physical systems with emergent collective computational abilities", Proceedings of the National Academy of Sciences of the USA, vol. 79 no. 8 pp. 2554–2558, April 1982.</p>
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<p>[3] J. J. Hopfield. ''Neural networks and physical systems with emergent collective computational abilities''. Proceedings of the National Academy of Sciences of the USA, vol. 79 no. 8 pp. 2554–2558, April 1982.</p>
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Latest revision as of 03:50, 27 September 2012