Stochastic Simulations of Genetic Regulatory Networks: The Genetic Toggle Switch
Stochastic Simulations
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Transcript of Stochastic Simulations
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Stochastic Simulations
Monday, 9/9/2002
Monte Carlo simulations are generally concerned with large series of computer experiments using uncorrelated random numbers.
Explore order out of randomness
•Random sampling•Fractoemission•Diffusion•Polymer•Growth model
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Hit-or-Miss Random Sampling
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π ⋅12
22 =nhitnall
π =4nhitnall
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Buffon’s Needle
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Fracto-emission
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Fracto-emission Measuring System
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Zigzag Crack Profile Model
Fracto-emission particles bounce at the irregular surfaces.
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Longtime Decay of theFracto-emission Intensity
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Random Walk
Haphazrad paths on a lattice
A drop of ink through water.
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One Dimensional Random Walk
http://polymer.bu.edu/java/java/1drw/1drwapplet.html
Wandering ant
Try and extract an equation from the plot relating the mean squared distance to the step number.
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Question
How do the answers change is the probability is p (!= 1/2) to move right and 1-p to move left (a forward- or reverse-biased motion)?
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Diffusion
Screen shots of the trajectory of 500 random walkers, started together at the center.
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Extension of Random WalkThis model is a two-dimensional extension of a random walk. Displayed is the territory covered by 500 random walkers. As the number of walkers increases the resulting interface becomes more smooth.
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Different kinds of random walks on a square lattice
Random Walk (RW): the walker may cross the walk in an infinite number of times with no cost.
Self-Avoiding Walk (SAW): the walker dies when attempting to intersect a portion of the already completed walk.
Growing Self-Avoiding Walk (GSAW): the process proceeds at first as for SAWs, but a walker senses a ‘trap’ and chooses instead between the remaining
‘safe’ directions so that it can cancontinue to grow.
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Polymer Model
Bond lengths of polymers tend to be rather fixed as do bond angles. Thus, as a more computationally friendly model we may construct a polymer which is made up of bonds which connect nearest neighbor sites (monomers) on a lattice.
Schematic model for polyethylene
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Polymers as Long
Molecular Chains
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Self-Avoiding Random Walk
Mean square distance of gyration of a linear polymer molecule consists of N monomer unites has the leading asymptotic behavior
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Rg2 =AN2ν
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Diffusion Limited Aggregation (DLA)
A seed is placed at the center of the box. A point is chosen at random in the box, excluding a zone around the cluster. A particle then random walks from this point until it either sticks to the cluster or is lost from the box.
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DLA Growth Model
http://apricot.polyu.edu.hk/~lam/dla/dla.html
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Thermodynamic Force DrivenSelf Assembly
How to grow desired fine nanoscale structures by pre-patterning some coarse structures.
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Monte Carlo vs.Kinetic Monte Carlo
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pMC =exp−E1 −E2
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pkMC =exp−E1 −E2
kBT
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