Difference between revisions of "Team:Kent/Modeling"

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<p> Modeling is important as it allows us to describe the system mathematically. If we change some of the parameters in our system we can see how this will affect the system, this is especially important when the some of the parameters are unknown. The main aim of our model is to demonstrate the production of our nanowires in an interactive and interesting way. </p>
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Modeling, i.e. the mathematical description of a physical system, is an important aspect of the scientific method as it allows us to quantitatively understand the system and to make predictions. Specifically, a model helps understanding how changes in any parameter affect the system; this is especially important when the relevance of some parameters is unknown. We chose to develop a simulation model of the system studied in our project, which consists of a self-assembling biological structure that we thought would be exciting and informative to quantify and visualize in an interactive way. We made the computer code of the model publicly available, so that other teams can play around and build upon it.
  
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<p align="center"> More to come soon... </p>
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We used the Monte Carlo method to simulate the diffusion [3] of monomers inside an E. coli cell, their transit through the cell membrane and the production of amyloid nanowires outside the cell. A typical E coli cell has a length, l=2μm and a diameter, d=1μm. We take a small observation cube, which is a portion of both the inside of the cell and the bulk outside the cell; we can extrapolate this to describe the whole system. Simulating just a portion has the advantage of requiring less computational power.
  
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The monomers initially start at the bottom of the observation volume and are allowed to stochastically diffuse inside the cell and go through the cell membrane. The binding site seeds are located on top of the cells membrane and when a monomer gets close enough to the binding site it may bind and form a link in the chain. Over time the chains can grow to lengths in the range of 60nm to 100μm [1][5][10], these chains are not necessarily straight and persistence length parameter dictates the angles at which particles can bind. The monomer can only interact with the monomer attached to the chain, i.e. there is no branching.
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<p>
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Modeling is important as it allows us to describe the system mathematically. If we change some of the parameters we can see how this will affect the system, this is especially important when the some of the parameters are unknown. We chose to create a simulation as our project dealt with a self-assembling structure that we thought would be exciting to visualize in an interactive way. We have included our code so that other teams can play around and build upon it.
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Periodic and reflective boundary conditions are applied in the model. When a particle leaves through the side of the observation volume, we can assume that another particle enters through the other side. When a particle reaches the bottom of the observation volume we can assume that the particle is reflected. This allows us to reproduce behaviour similar to the bulk.  
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<br><br>
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We used the Monte Carlo method to simulate the stochastic diffusion [3] of monomers from inside a cell and how this leads to the production of amyloid nanowires. A typical E coli cell has a length, l=2μm and a diameter, d=1μm. We take a small observation cube, which looks at both the cell and the bulk outside of the cell; we can extrapolate this to describe the whole system. This has the advantage of requiring less computational power.
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The number of monomers in the system is not necessarily constant; monomers can be created and degraded. However, Hall [8] (2003) argued with the two extreme cases that either; the monomer are generated and degraded on a time scale much slower than amyloid growth so the number of particles are constant; or that the amyloid growth occurs on a time scale much slower than monomer degradation so we can refer to the free concentration of the monomer as constant.  
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<br><br>
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The monomers initially start at the bottom of the observation volume and are allowed to stochastically diffuse. The binding site seeds are located on top of the cells membrane and when a monomer gets close enough to the binding site it may bind and form a link in the chain. Over time the chains can grow to lengths in the range of 60nm to 100μm [1][5][10], these chains are not necessarily straight and persistence length parameter dictates the angles at which particles can bind. The monomer can only interact with the monomer attached to the chain, i.e. there is no branching.
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The Monte Carlo simulation was implemented in Matlab and visualized using Visual Molecular Dynamics (VMD).
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<br><br>
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The system has a periodic potential. When a particle leaves through the side of the observation volume, we can assume that another particle enters through the other side. When a particle reaches the bottom of the observation volume we can assume that the particle is reflected.
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<br><br>
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The number of monomers in the system is not necessarily constant; monomers can be created and degraded. However, Hall [8] (2003) argued with the two extreme cases that either; the monomer occur on a time scale much slower than amyloid growth so the number of particles are constant; or that the amyloid growth occur on a time scale much slower than monomer degradation so we can refer to the free concentration of the monomer as constant.  
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<br><br>
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The Monte Carlo method was simulated in Matlab and visualized using Visual Molecular Dynamics (VMD).
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     <th> Variable </th> <th> Description </th> <th> Value </th>
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     <th> Variable </th> <th> Description </th> <th> Value </th> <th> Reference </th>
 
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     <th> \(L_x\)</th><th> Length of the observation box volume </th><th> \(0.4 \mu m\) </th>
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     <th> \(L_x\)</th><th> Length of the observation box volume </th><th> \(0.4 \mu m\) </th> <th> </th>
 
<tr>
 
<tr>
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     <th> \(L_y\)</th><th> Depth of the observation box volume </th><th> \(0.4 \mu m\) </th>
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     <th> \(L_y\)</th><th> Depth of the observation box volume </th><th> \(0.4 \mu m\) </th><th> </th>
 
<tr>
 
<tr>
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     <th> \(L_z\) </th><th>Height of the cell in observation volume </th><th>\(0.5 \mu m\) </th>
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     <th> \(L_z\) </th><th>Height of the cell in observation volume </th><th>\(0.5 \mu m\) </th> <th></th>
 
<tr>
 
<tr>
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     <th> \(N_b\) </th> <th> Number of binding sites </th> <th> 5 </th>
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     <th> \(N_b\) </th> <th> Number of binding sites </th> <th> 5 </th><th> </th>
 
<tr>
 
<tr>
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     <th> \(N(0)\)</th><th> Initial number of particles </th><th> 1000 </th>
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     <th> \(N(0)\)</th><th> Initial number of particles </th><th> 2000 </th><th> </th>
 
<tr>
 
<tr>
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     <th> \(\Delta t\)</th><th> Length of each timestep </th><th> \(\frac{1}{2000} s\) </th>
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     <th> \(\Delta t\)</th><th> Length of each timestep </th><th> \(\frac{1}{2000} s\) </th> <th> </th>
 
<tr>
 
<tr>
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     <th> \(\lambda \) </th><th> Persistence Length </th><th> \(1000 \mu m\) </th>
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     <th> \(\lambda \) </th><th> Persistence Length </th><th> \(23 \mu m\) </th> <th> [7][11] </th>
 
<tr>
 
<tr>
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    <th> \(r\) </th><th> Monomer radius </th><th> \( 0.0015 \mu m\) </th> <th> </th>
 
<tr>
 
<tr>
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     <th> \(r\) </th><th> Monomer radius </th><th> \( 0.0015 \mu m\) </th>
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     <th> \(r_b \) </th><th> Binding radius </th><th> \(0.0030 \mu m \) </th> <th></th>
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    <th> \(r_b \) </th><th> Binding radius </th><th> \(0.0030 \mu m \) </th>
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<tr>
 
<tr>
 
     <th> \(D_{chain} \) </th><th> Diffusion coefficient of the chain after the chain has detached </th><th> \(0.0005 \mu m^2 /s \) </th>
 
     <th> \(D_{chain} \) </th><th> Diffusion coefficient of the chain after the chain has detached </th><th> \(0.0005 \mu m^2 /s \) </th>
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<table>
 
<table>
 
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     <th> Event </th> <th> Description </th> <th> Value </th>
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     <th> Event </th> <th> Description </th> <th> Value </th> <th> Reference </th>
 
<tr>
 
<tr>
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     <th> P(bind)</th><th> Probability that a monomer, within range, will attach to the end of a chain </th><th> 0.5 </th>
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     <th> \(P(bind) \)</th><th> Probability that a monomer, within range, will attach to the end of a chain </th><th> 0.5 </th><th></th>
 
<tr>
 
<tr>
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     <th> P(leave)</th><th> Probability that a monomer will leave the cell from the inside </th><th> 0.5 </th>
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     <th> \(P(leave) \)</th><th> Probability that a monomer will leave the cell from the inside </th><th> 0.5 </th> <th></th>
 
<tr>
 
<tr>
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     <th> P(spawn) </th><th> Probability that a monomer will be created at the end of each timestep </th><th>0.5 </th>
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     <th> \(P(spawn) \) </th><th> Probability that a monomer will be created at the end of each timestep </th><th>0.5 </th> <th> [8]</th>
 
<tr>
 
<tr>
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     <th> P(decay) </th> <th> Probability that any given monomer will decay at each timestep </th> <th> 0.001 </th>
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     <th> \(P(decay) \) </th> <th> Probability that any given monomer will decay at each timestep </th> <th> 0.001 </th> <th> [8] </th>
 
<tr>
 
<tr>
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     <th> P(monomer detach)</th><th> Probability that the last monomer on a chain will detach </th><th> 0.001 </th>
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     <th> \(P(es) \)</th><th> Probability that the last monomer on a chain will detach </th><th> 0.001 </th> <th> [8][12] </th>
 
<tr>
 
<tr>
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     <th> P(anchor detach)</th><th> Probability that the chain will detach from the membrane </th><th> 0.0001 </th>
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     <th> \(P(anchor detach) \)</th><th> Probability that the chain will detach from the membrane </th><th> 0.0001 </th> <th> </th>
 
<tr>
 
<tr>
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     <th> P(fragment) </th><th> Probability that a chain of j monomers will fragment </th><th> \( (j-1)*0.0000.1 \) </th>
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     <th> \(P(fragment) \) </th><th> Probability that a chain of j monomers will fragment </th><th> \( (j-1)*0.0000.1 \) </th> <th> [8][12] </th>
 
<tr>
 
<tr>
 
</table>
 
</table>
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</p>
 
</p>
  
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<a name="c2"></a><h3 align="center"> Observation Volume </h3>
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<a name="c2"></a><h3 align="center"> Observation box volume and area </h3>
 
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We have taken a small proportion of the cell and the bulk outside of the cell and extrapolated it to represent the whole system.
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We have used a small observation box to represent the whole cell. It is important to know what proportion of the cell is represented so that the results can be extrapolated properly.
 
</p><br><br>
 
</p><br><br>
 
<p>The typical length of an E coli cell is \(L \approx 2 \mu m\) and the typical diameter, \(d \approx 1 \mu m\). If we consider the cell to be composed of a cylinder of length \(l=L-d\) and two hemispheres at each end then the volume of the sphere is:</p>
 
<p>The typical length of an E coli cell is \(L \approx 2 \mu m\) and the typical diameter, \(d \approx 1 \mu m\). If we consider the cell to be composed of a cylinder of length \(l=L-d\) and two hemispheres at each end then the volume of the sphere is:</p>
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<a name="c3"></a><h3 align="center"> M</h3>
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<a name="c3"></a><h3 align="Stochastic Brownian motion"> M</h3>
  
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<p> The displacement each time step in the x,y "and " z directions is given by a simple form of the Langevin equation. This model assumes that there are no deterministic forces acting on the monomers, i.e. pure diffusion [3][4]</p>
 
\[ \Delta x = \xi \sqrt{2D \Delta t} \]
 
\[ \Delta x = \xi \sqrt{2D \Delta t} \]
  
 
\[D(z) = c + \tanh(k(z - z_0)) \]
 
\[D(z) = c + \tanh(k(z - z_0)) \]
  
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<a name="c4"></a><h3 align="center"> Boundary conditions   </h3>
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<p> We make the assumption that if a monomer leaves through the side of the observation volume  that another particle will enter through the other side. We also assume that if a monomer hits the bottom of the observation volume that it will be reflected. In the following equations, \(x^' \), \(y^' \) and \(z' \) denote the respective x,y and z coordinates of the particles after the application of boundary conditions: </p>
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<a name="c4"></a><h3 align="center"> E   </h3>
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\[ x' = \begin{cases}1 & -1 \le x < 0 \\ \frac{1}{2} & x = 0 \\ 1 - x^2 & \text{otherwise} \end{cases} \]
 
\[ x' = \begin{cases}1 & -1 \le x < 0 \\ \frac{1}{2} & x = 0 \\ 1 - x^2 & \text{otherwise} \end{cases} \]
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\[z' = -z \]
 
\[z' = -z \]
  
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<p> If a particle on the inside moves to a position that has a height greater than the height of the cell then it has a probability \(P(leave) \) of successfully leaving the cell. Likewise if a particle on the outside of the cell moves to a position that has a height less than the height of the cell then it has a probability \(P(enter) \)of successfully entering the cell. In both of these scenarios, if the particle fails to pass through the membrane then they are relocated in the z position by:</p>
 
\[z' = 2 L_z - z \]
 
\[z' = 2 L_z - z \]
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<a name="c5"></a><h3 align="center"> S  </h3>
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<a name="c5"></a><h3 align="Length of the chain"> S  </h3>
  
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<p> <strong> Nucleation </strong> </p>
 
\[ A + B \overset{k}{\rightarrow} AB\]
 
\[ A + B \overset{k}{\rightarrow} AB\]
  
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<p> <strong> Elongation </strong> </p>
 
\[ A_{j}B \overset{k_{+}}{\rightarrow} A_{j + 1}B \]
 
\[ A_{j}B \overset{k_{+}}{\rightarrow} A_{j + 1}B \]
  
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<p> <strong> Fragmentation </strong> </p>
 
\[ A_{j}B \overset{k_{-}}{\rightarrow} A_{i}B + A_{j-i} \]
 
\[ A_{j}B \overset{k_{-}}{\rightarrow} A_{i}B + A_{j-i} \]
  
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\[ \frac{\partial f(t,k)}{\partial t} = 2m(t) K_{+} f(t,j-1) - 2m(t)k_{+}f(t,j_ - k_{-}(j-1)f(t,j) + 2k_{-} \sum_{i=j+1}^\infty f(t,i) + k_{n}m(t)^{n_{c}} \delta _{j,n_{c}} (1) \]
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<p> Where \(A \) is a monomer, \(B \) is the binding site; \(AB \) is a monomer-binding site complex and \(A_j B \) denotes a chain of \(j \) monomers attached to a binding site. For the reaction rates, a least squares fit from both Hall [8](2004) and Xue [12](2013), revealed the elongation rate to be of order \(k_+~10^5 mol^{-1} s^{-1} \) and the fragmentation rate to be of order \(k_-~10^{-8} s^{-1} \), the rate of elongation is much higher than that of fragmentation as \( \frac{k_+}{k_}-=10^{13} mol^{-1} \). </p>
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\[P = \sum_{j=n_c}^\infty f(t,j) \]
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\[M = \sum_{j=n_c}^\infty j * f(t,j) \]
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\[ <\cos (\alpha)> = \exp \big(- \frac{L}{\lambda } \big) = \exp \big(- \frac{jl}{\lambda} \big) \]
 
\[ <\cos (\alpha)> = \exp \big(- \frac{L}{\lambda } \big) = \exp \big(- \frac{jl}{\lambda} \big) \]
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Revision as of 16:42, 14 September 2015


iGEM Kent 2015


Modeling

Modeling, i.e. the mathematical description of a physical system, is an important aspect of the scientific method as it allows us to quantitatively understand the system and to make predictions. Specifically, a model helps understanding how changes in any parameter affect the system; this is especially important when the relevance of some parameters is unknown. We chose to develop a simulation model of the system studied in our project, which consists of a self-assembling biological structure that we thought would be exciting and informative to quantify and visualize in an interactive way. We made the computer code of the model publicly available, so that other teams can play around and build upon it.