Team:LCG-UNAM-Mexico/Simulation

 
 

 
With the aim of predicting the behavior of the system, the biochemical reactions were implemented in the SimBiology package of MATLAB, using the previously defined parameters. Simulations were run for different values of the initial concentration of AHL and Ni_total (Ni_int + Ni_ext) which are the metabolites that we can directly manipulate in our experiments. A parameter scan was also run for some parameters to understand their influence on the system.
In order to gain insights into the system dynamics to elucidate the conditions needed to get the desired behavior, we performed a series of analyses on it: Sensitivity analysis allowed us to identify critical parameters that needed to be defined on the most stringent way. Basis for the (right) null and left null space were calculated to obtain information about the general network behavior. Steady-states were calculated by numerical integration of the non-linear ODEs system. Finally the Jacobian of the system was calculated around the steady-states. All simulations and analyses were implemented and performed on MATLAB.

Simulation and parameter scan

Describir el comportamiento que queremos ver y por qué.
Incluir las gráficas de parameter scan, la gráfica de la vida, y el escaneo con el que definimos algunas constantes

Sensitivity analysis

Definir brevemente de que se trata, mostrar análisis a diferentes tiempos. Señalar los resultados que esperábamos y los que no

Stoichiometric matrix

Definir la información que contiene la matriz estequimétrica.
Definir los espacios nulos (link a wikipedia o matworld?)
Presentar las bases calculadas y una interpretación concisa

Steady-states

The steady state of a system is defined as a nonequilibrium state through which matter is flowing and in which all components remain at a constant concentration^ref. We defined the time taken to reach the steady state as the time when the variation in the concentrations of all components cease to be biologically relevant (here defined as variation in less than one molecule). We can have two initial states in our system:

With no AHL. In this case, the only reaction that takes place is the uptake and efflux of nickel, which instantaneously reaches the steady state.
With AHL. Here, all the regulatory steps in the cascade take place and the steady state is reached when all AHL has been degraded and the signal has faded.

In both cases, the steady state reached is the same. The reason for this is that the system is a regulatory cascade and does not have any feedback loops, so it is expected to have only one steady state. Here we present the simulation in 100,000 seconds, you can click on in to see a larger version:

The concentrations at which it is reached are the following:

The range of AHL concentration in this analysis was chosen from 30 (when we start to see a measurable response) to 10,000. We can observe a linear dependence with higher AHL concentrations. The extent of RcnA repression can be adjusted at will as long as the extracellular concentration of nickel is not damaging to the cell.

Jacobian

The Jacobian of a system is defined as the matrix of first order partial derivatives, ad it represents the best linear approximation to a function at a given point. In biochemical networks, the Jacobian can be defined for metabolites(Jx) or fluxes(Jv):

Where S is the stoichiometric matrix and G is the gradient matrix. S defines the structure of the network and has the stoichiometric coefficients of all reactions which are represented by the rows of S while metabolites are represented by columns. G is the matrix of first order derivatives of fluxes with respect to species concentrations:

This formal representation of the Jacobian formalizes the relation between the topology of the network, and its biophysical and kinetic characteristics (REFERENCIAS DE SUR UNO Y DOS).

For our system, we first obtained S which is rank deficient: its rank is 7 and it has 13 rows, this difference is explained by the moieties of the system. We reduced S to make it congruent with its rank by eliminating the rows corresponding to: AiiA, LuxR, pcI, p, Ni-ext and Unk. Then we constructed G as previously defined; the partial derivatives are calculated only with respect to the species that remained in S; the state at which G is calculated is the steady-state.

Both Jacobians were calculated and they were decomposed through similarity transformation of their eigenvalues and eigenvectos to obtain the modal matrices. This matrices represent the formation of molecular ‘pools’ on the system and the negative inverse of the eigenvalus represent the time scales at which this interactions occur. The Jacobians and modal matrices can be downloaded on Excel (.xls) format if you click here.

The values in the tables show that the nickel response is very fast, in the order of 6.03e-8 s, which is what we need to make this system and efficient transcriptional indicator. On the other hand, there is a pool of RcnA that forms immediately and another one that takes much more time to form.

REFERENCIAS DE SUR
Jamshidi N and Palsson (2008) Formulating genome-scale kinetic models in the post-genome era Molecular Systems Biology 4:171.
Jamshidi N and Palsson (2008) Top-Down analysis of temporal hierarchy in biochemical reactions networks PLoS Comput Biol 4(9).