Protein Amyloid Aggregation: Numerical Simulation and the Monte Carlo Method Anna Woodard Diego Conc

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Protein Amyloid AggregationNumerical Simulation
and the Monte Carlo MethodAnna WoodardDiego
Concha Advisor Dr. Shaohua Xu
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• The Problem
• Colloidal Spheres attract despite having a
  positive net charge. Why? How? What can prevent
  it?
• In vitro experiments are definitive, but time
  consuming and expensive.
• We need to know how a vast array of variables
  will affect the aggregation process
• Protein concentration
• Electrolytes (concentration, valence, etc.)
• Small molecules and potential drugs
• Temperature, pH, etc.
• We cannot test all the possibilities

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• We can simulate the process - with a model
• What is a model?
• A mathematical description sufficiently detailed
  to predict the response of a real process under a
  range of conditions
• Simplifying assumptions are unavoidable
• Why do we need a model?
• To show that our understanding of the process is
  correct
• To predict the effects of variations in the
  system
• How do we build a model?
• Experiments generate observations
• Observations are used to form a model
• The model is used to generate many predictions
• Some predictions are verified by new experiments

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• Models for colloidal interaction
• DLVO Theory
• Preceded computers (1950's) closed form
  integration
• Very successful in predicting colloidal behaviour
• As more factors are considered, the equations
  become very complex with many arbitrary constants
• W(D)(64pkTR?8?/?2)e-?D-AR/6D
• RPhysicists hate arbitrary constants
• Model should predict experiments, not just depend
  on them

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• We used Maple to calculate the forces between
  nucleation units under the DLVO approximation.
• Neglecting ion shielding of dipoles, the
  attraction produced by only a few peptide bond
  dipoles (10-30 Coulomb-meters.) in each
  nucleation unit would be sufficient to completely
  overcome the energy barrier caused by coulomb
  repulsion Van der Waals forces.
• Spherical chicken in a vacuum
• This is only a crude approximation but it
  suggests that even a moderate degree of peptide
  bond alignment would permit linear aggregation.
• DVLO was not specifically derived to deal with
  charged dipoles.

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• Monte Carlo Method
• Name refers to generation of random numbers by a
  roulette wheel
• Requires only that we can calculate the energy
  for a given state
• Step 1 Generate a large number of possible
  states at random
• Step 2 Pick the one with the best value for the
  parameter
• In this case, the arrangement of ions with the
  lowest total energy will be the stable state
• Problem with basic Monte Carlo method
  Dimensionality is too high
• Dimensionality spacial dimensionsnumber of
  molecules
• With a 1000A box and 100 molecules, sample space
  
• 1000(3100) (1 with 303 zeros) total states
• Cannot evaluate a significant part of the sample
  space!

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• Solution Metropolis Algorithm
• Start with a single arrangement
• Randomly move one ion/atom
• If there is a hard-sphere overlap, reject the new
  state
• Calculate the change in energy based on the
  change in distance to other ions
• If energy is lower, accept the new state
• If energy is higher, usually reject, sometimes
  accept
• a statistically possible random walk thermal
  fluctuation
• Repeat until energy stops decreasing
• System now at equilibrium
• We don't need to know the forces only the
  energy!

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• Only two stable patterns of linear aggregation
• Depend on length and width
• Spherical or short fat dipoles nose-to-tail
  chain
• Long, thin dipolar particles side-by-side with
  adjacent poles reversed
• With some species, short nose-to-tail chains will
  behave like long dipoles and aggregate
  side-by-side with reversed polarity to form
  thicker chains
• Doesn't happen initially in proteins (due to
  charge?) but multiple-ribbon appearance in some
  cases could indicate multiple chains with
  reversed dipoles
• Why dont the fibers bend?

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Striolo (2002) determined that DVLO was not
accurate for charged dipolar colloids (even with
complex equation that attempts to allow for
dipoles) because of the dipoles' effect on ion
concentrations between the particles
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• Do dipolar molecules aggregate?

methane (nonpolar)
methane (strongly dipolar)
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Effects of Charge and Van Der Walls Forces
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• Outlook Water, ions, etc. . .
• Simulate aggregation of multiple nucleation units
• Calculate required dipole for aggregation
• Ultimately, actual internal macromolecular
  structure

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This is a complex puzzle - but molecular
modeling is part of the solution Thank You