Team:British Columbia/Modeling

From 2013.igem.org

(Difference between revisions)

Revision as of 20:42, 31 August 2013

Reaction Kinetics: - Modelling the production of cinnamaldehyde, vanillin and caffeine production

Population dynamics: - Modelling the population of a bacteria in a batch reactor with phage dependency

\begin{align} \frac{dX}{dt} = \frac{dX_u}{dt} + \frac{dX_i}{dt} + \Gamma \end{align} Where $\frac{dX}{dt}$ is rate of change in bacteria population over time, $\frac{dX_u}{dt}$ and $\frac{dX_i}{dt}$ are the rates of change in the uninfected and infected populations respectively. $\Gamma$ is defined as a stepwise function. \begin{align} \Gamma &= - E(X_i), \ during \ phage \ caused \ lysis\\ &= 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise \end{align} Where $E(X_i)$ is the statistically expected infected bacterial populations. We define the rate of change in population in () and (). \begin{align} \frac{dX_u}{dt} = \mu_u + k_d \end{align}

@@ Line 9: / Line 9: @@
 - Modelling the population of a bacteria in a batch reactor with phage dependency
-We've decided to use Monod substrate limited growth kinetics for to model population dynamics.
 \begin{align}
-\mu = \mu_{max} \frac{S(t)}{K_s + S(t)}
+\frac{dX}{dt} = \frac{dX_u}{dt} + \frac{dX_i}{dt} + \Gamma
 \end{align}
+Where $\frac{dX}{dt}$ is rate of change in bacteria population over time, $\frac{dX_u}{dt}$ and $\frac{dX_i}{dt}$ are the rates of change in the uninfected and infected populations respectively. $\Gamma$ is defined as a stepwise function.
-Where $\mu$ growth rate in units hr$^{-1}$, $\mu_{max}$ is the maximum growth rate, $K_s$ is the amount of substrate remaining when $\frac{\mu}{\mu_{max}} = \frac{1}{2}$, and $S(t)$ is the substrate remaining at a given time.
-Since growth rate is defined as:
 \begin{align}
-\mu \equiv \frac{1}{X(t)} \frac{dX}{dt} + k_d
+\Gamma &= - E(X_i),  \ during \ phage \ caused \ lysis\\
+&= 0, \ \ \ \ \ \ \ \ \ \ \ \ \ \  otherwise
+\end{align}
+Where $E(X_i)$ is the statistically expected infected bacterial populations. We define the rate of change in population in () and ().
+\begin{align}
+\frac{dX_u}{dt} = \mu_u + k_d
 \end{align}
-Substituting (1) into (2) and rearranging we arrive at a description of rate of population change:
-Combined Reaction Kinetics/Population Dynamics:
-- Creating a function for product produced as a function of phage added.