Team:Oxford/Test/Modeling2
Modeling
Introduction
Modeling allows us to observe what happens to a system when you change the parameters that describe it. By quantifying the system’s response to these variables, we can aid the Biochemists in their design of their biological system. By repeating the modeling, data-fitting and refining schedule we approach a sufficient description of our required design.
Our modeling is made up of three pieces - gene expression, diffusion and biofilm modeling. By modeling gene expression, we can inform the lab team which parameters they can choose freely - and which ones they must pick carefully - to maximise the output of our product. The gene copy number is one example of these parameters. By modeling diffusion, we can incorporate into our previous model the effect of our protein being separated from the biofilm it is trying to interact with. If this is a large effect, the lab team must change the design of their system. Finally, by modeling the formation of a biofilm and its interaction with our product, we can decide upon the required concentrations of product we need to clear the urinary tract infection. In all cases we use both deterministic and stochastic models to investigate our system.
Gene Expression
Deterministic
A deterministic model evolves according to ordinary or partial differential equations. We model the following system:
\[\overset{\alpha_{1}}{\rightarrow}Arab\quad\overset{\alpha_{5}}{\rightarrow}AraC\] \[Arab+AraC\stackrel[\alpha_{3}]{\alpha_{2}}{\rightleftharpoons}Arab:AraC\] \[\overset{K}{\rightarrow}mRNA\overset{\alpha_{4}}{\rightarrow}P\] \[Arab\overset{\gamma_{1}}{\rightarrow}\phi\quad AraC\overset{\gamma_{2}}{\rightarrow}\phi\quad mRNA\overset{\gamma_{3}}{\rightarrow}\phi\quad P\overset{\gamma_{4}}{\rightarrow}\phi\]Using the following deterministic equations [4]:
\[\dfrac{d\left[Arab\right]}{dt}=\alpha_{1}+\alpha_{2}\left[Arab\right]\left[AraC\right]-\alpha_{3}\left[Arab:AraC\right]-\gamma_{1}\left[Arab\right]\] \[\dfrac{d\left[Arab:AraC\right]}{dt}=\alpha_{3}\left[Arab:AraC\right]-\alpha_{2}\left[Arab\right]\left[AraC\right]\] \[\dfrac{d\left[AraC\right]}{dt}=\alpha_{5}+\alpha_{2}\left[Arab\right]\left[AraC\right]-\alpha_{3}\left[Arab:AraC\right]-\gamma_{2}\left[AraC\right]\] \[\dfrac{d[AraC]}{dt}=\alpha_{5}+\alpha_{2}[Arab][AraC]-\alpha_{3}[Arab:AraC]-\gamma_{2}[AraC]\] \[\dfrac{d[mRNA]}{dt}=K_{max}\dfrac{[Arab:AraC]^{n}}{K_{half}^{n}+[Arab:AraC]^{n}}-\gamma_{3}[mRNA]\] \[\dfrac{d\left[P\right]}{dt}=\alpha_{4}\left[mRNA\right]-\gamma_{4}\left[P\right]\]To evaluate these parameters, we will fit the data created by the biochemists. We allow a tolerance of a factor of ten either side of our initial guess for each parameter. The fitted parameters will be given in the fourth column and will be found using the MATLAB function fmincon. To solve the equations numerically we use the MATLAB equation solver ode15s.
Stochastic
A stochastic model is anything which includes some random element, and is useful for modeling small systems or those with some uncertainty. We produced a video tutorial on how to model gene expression networks stochastically using Gillespie's algorithm. We summarise the Gillespie algorithm in below [2, 4].
Diffusion
Deterministic
Stochastic
Random Walk
Random Velocity
Compartmental diffusion; Gillespie
Comparison of methods
All of the methods provide similar levels of accuracy (depending on the size of the compartments used in the Gillespie model) and so we choose based on the simplicity of implementation. The random walk method is generally the fastest and easiest to code, so we choose this one.
The challenge
Since we are modelling diffusion in the catheter, the topology of the system is a ring, neglecting edge effects of the cylinder our bacteria will be placed in. To do this we use a random walk model in two dimensions. We confine our particles to a ring of a fixed radius and initially coat the ring with particles.
Modeling Tutorial
Deterministic
Stochastic
The simplest way to add an element of 'randomness' into an equation is to add a random term into the differential equation. This turns it into a stochastic differential equation, such as:
This determines the evolution of a particle travelling at a speed c, with a random number r added into the mix just for fun. We choose r to be normally distributed with a mean value of 0 and a variance of, say, 1. We evaluate the position of the particle at multiple timesteps (with some small time dt separating them). Below we plot this in MATLAB.
We can do better than this when dealing with systems such as our example in the previous section. We use the Gillespie algorithm to find:
- The time taken between reactions
- The reaction that took place in that time.
as neither of these could be determined by our initial method [2, 4].
1) To find the time taken, we make an assumption about the form of the probability distribution that the time between reactions holds. It makes sense to think of it as a falling exponential, such that reactions more often happen quickly rather than slowly, and that the more probable a reaction is, the shorter the time between subsequent reactions. This naturally leads to the form
2) To decide which reaction took place, we need the probabilities of each reaction occuring being proportional to the ratio of their propensities. We split up the domain of 0 to 1 into the normalised ratios of the propensities of our problem, recalling the propensities we calculated. We could have, in this case, selected the backwards reaction B to A. We can generalise this to the form given in the Gillespie Algorithm above.
Finite Difference Models
References