Team:Kent/Modeling
Modeling
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.
More to come soon...
Contents
VariablesChapter 2
Chapter 3
Chapter 4
Chapter 5
References
Variables
| Variable | Description | Value |
|---|---|---|
| \(L_x\) | Length of the observation box volume | \(0.4 \mu m\) |
| \(L_y\) | Depth of the observation box volume | \(0.4 \mu m\) |
| \(L_z\) | Height of the cell in observation volume | \(0.5 \mu m\) |
| \(N_b\) | Number of binding sites | 5 |
| \(N(0)\) | Initial number of particles | 1000 |
| \(\Delta t\) | Length of each timestep | \(\frac{1}{2000} s\) |
| \(\lambda \) | Persistence Length | \(1000 \mu m\) |
| \(r\) | Monomer radius | \( 0.0015 \mu m\) |
| \(r_b \) | Binding radius | \(0.0030 \mu m \) |
| \(D_{chain} \) | Diffusion coefficient of the chain after the chain has detached | \(0.0005 \mu m^2 /s \) |
| Event | Description | Value |
|---|---|---|
| P(bind) | Probability that a monomer, within range, will attach to the end of a chain | 0.5 |
| P(leave) | Probability that a monomer will leave the cell from the inside | 0.5 |
| P(spawn) | Probability that a monomer will be created at the end of each timestep | 0.5 |
| P(decay) | Probability that any given monomer will decay at each timestep | 0.001 |
| P(monomer detach) | Probability that the last monomer on a chain will detach | 0.001 |
| P(anchor detach) | Probability that the chain will detach from the membrane | 0.0001 |
| P(fragment) | Probability that a chain of j monomers will fragment | \( (j-1)*0.0000.1 \) |
Observation Volume
We have taken a small proportion of the cell and the bulk outside of the cell and extrapolated it to represent the whole system.
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:
\[V_{cell} = V_{cylinder} + V_{sphere}\]
\[V_{cell} = \pi \big(\frac{d}{2}\big)^2 (L-d)+ \frac{4}{3} \pi \big(\frac{d}{2}\big)^3 = 1.309 (\mu m)^3\]
The volume of the cell inside the observation box is
\[V_{ob} = L_x L_y L_z = 0.08(\mu m)^3\]
From this we can work out the proportion of the volume of cell inside the observation box
\[\frac{V_{ob}}{V_{cell} } = 6.112%\]
Likewise for the surface area of the cell
\[A_{cell} = A_{cylinder} + A_{sphere}\]
\[A_cell = \pi d(L-d) + 4 \pi (\frac{d}{2})^2 = 6.28 (\mu m)^2 \]
The proportion of the cell's surface area inside the observation box is:
\[\frac{A_{ob}}{A_{cell}} = 2.54% \]
M
\[ \Delta x = \xi \sqrt{2D \Delta t} \] \[D(z) = c + \tanh(k(z - z_0)) \]Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
E
\[ x' = \begin{cases}1 & -1 \le x < 0 \\ \frac{1}{2} & x = 0 \\ 1 - x^2 & \text{otherwise} \end{cases} \] \[ y' = \begin{cases}1 & -1 \le x < 0 \\ \frac{1}{2} & x = 0 \\ 1 - x^2 & \text{otherwise} \end{cases} \] \[z' = -z \] \[z' = 2 L_z - z \]Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
S
\[ A + B \overset{k}{\rightarrow} AB\] \[ A_{j}B \overset{k_{+}}{\rightarrow} A_{j + 1}B \] \[ A_{j}B \overset{k_{-}}{\rightarrow} A_{i}B + A_{j-i} \] \[ \frac{\partial f(t,k)}{\partial t} = 2m(t) K_{+} f(t,j-1) - 2m(t)k_{+}f(t,j_ - k_{-}(j-1)f(t,j) + 2k_{-} \sum_{i=j+1}^\infty f(t,i) + k_{n}m(t)^{n_{c}} \delta _{j,n_{c}} (1) \] \[P = \sum_{j=n_c}^\infty f(t,j) \] \[M = \sum_{j=n_c}^\infty j * f(t,j) \] \[ <\cos (\alpha)> = \exp \big(- \frac{L}{\lambda } \big) = \exp \big(- \frac{jl}{\lambda} \big) \] \[ d_x = r \sin(\theta) \cos(\phi) \] \[ d_y = r \sin(\theta) \sin (\phi) \] \[ d_z = r \cos (\phi) \] \[ x = x_c + r \sin )\theta + \Delta \theta ) \cos (\phi + \Delta \phi \Delta t) \] \[ y = y_c + r \sin (\theta + \Delta \theta) \sin (\phi + \Delta \phi \Delta t) \] \[ z = z_c + r \cos (\theta + \Delta \theta ) \]Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
Our Matlab code can be found here: Kent15_Model.m
References
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[3] Philipse, A. P. (2011). Notes on Brownian Motion. Utrecht University, Debye Institute, Van’t Hoff Laboratory.
[4] Barlett, V. R., Hoyuelos, M., & Mártin, H. O. (2013). Monte Carlo simulation with fixed steplength for diffusion processes in nonhomogeneous media. Journal of Computational Physics, 239, 51-56.
[5] Xue, W. F., Homans, S. W., & Radford, S. E. (2008). Systematic analysis of nucleation-dependent polymerization reveals new insights into the mechanism of amyloid self-assembly. Proceedings of the National Academy of Sciences,105(26), 8926-8931.
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[7] Smith, J. F., Knowles, T. P., Dobson, C. M., MacPhee, C. E., & Welland, M. E. (2006). Characterization of the nanoscale properties of individual amyloid fibrils.Proceedings of the National Academy of Sciences, 103(43), 15806-15811.
[8] Hall, D., & Edskes, H. (2004). Silent prions lying in wait: a two-hit model of prion/amyloid formation and infection. Journal of molecular biology, 336(3), 775-786.
[9] Lomakin, A., Chung, D. S., Benedek, G. B., Kirschner, D. A., & Teplow, D. B. (1996). On the nucleation and growth of amyloid beta-protein fibrils: detection of nuclei and quantitation of rate constants. Proceedings of the National Academy of Sciences, 93(3), 1125-1129.
[10] Scheibel, T., Parthasarathy, R., Sawicki, G., Lin, X. M., Jaeger, H., & Lindquist, S. L. (2003). Conducting nanowires built by controlled self-assembly of amyloid fibers and selective metal deposition. Proceedings of the National Academy of Sciences, 100(8), 4527-4532.
[11] vandenAkker, C. C., Engel, M. F., Velikov, K. P., Bonn, M., & Koenderink, G. H. (2011). Morphology and persistence length of amyloid fibrils are correlated to peptide molecular structure. Journal of the American Chemical Society,133(45), 18030-18033.