Thermal Forces and Diffusion

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Thermal Forces and Diffusion

• Chapter 4
• Howard

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Thermal Forces and Diffusion

• Thermal Forces collisions with water and other
  molecules in surrounding fluid, force results
  from the change in momentum from the collision
• Amplitude proportional the temperature
• Brownian motion diffusion of a free particle
• Forces high relative to protein conf. changes
  allow to reach transition state
• Boltzmans Law probability of a molecule having
  a certain energy depends on the surrounding
  temperature

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Boltzmanns Law

• Probability, p, of finding the particle in state,
  i, that has energy, U, is the Boltzmann constant
  x temp (kT)
• pi 1/Z exp-Ui/kT, where Z constant
  ?iexp-Ui/kT
• The sum of all probabilities adds up to 1
• General Law applies to many different potential
  energies, kinetic energy, electronic or chemical
  state
• State of a particle defined by position and
  velocity of constituent atoms as well as their
  electronic states
• Probability of being in two states is given by
• p2/p1 exp(?U/kT)
• Equilibrium if Boltzmanns law holds, no net
  flux of particles
• Steady-state average properties do not change
  with time, can be a net flux of particles

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p2/p1 exp(?U/kT)
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Equipartition of Energy

• Can calculate average thermal energy of a
  molecule
• Molecule connected to a spring U(x) ½ kx2,
  calculate position
• Mean position or Mean squared position
• Boltzmanns law to calculate probability of
  finding molecule at position x

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Equipartition of Energy

• Average energy of a molecule ½kT
• Does not depend on stiffness only the temperature
• If average energy depends on the square of a
  parameter such as position or speed then average
  energy ltUgt ½kT
• Root-mean square speed, vrms in three dimension,
  (3kT/m)½
• Examples H20, 100 kDa Protein, Bacterium
• Can break down if thermal energy small holds
  for proteins at room temperature

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Diffusion as a random walk

• How the average concentration of a collection of
  molecules changes over time due to diffusion
• Movement from high concentration to low
  concentration . Rate is proportional to
  concentration gradient. J(x) -D dc/dx(x)
• Steady state no net change in concentration
  over time
• Probability of finding molecule at position x at
  time t , p (x,t)

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Einstein Relation

• Relating the diffusion coefficient to the drag
  coefficient
• D kT/?
• Estimate D, from size of particle and viscosity
  of the solution.
• Example diffusion of ions

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Solutions to the Diffusion Equation

• How quickly a molecules will diffuse in the cell
• Will diffusion be a viable method for a transport
  process
• Free diffusion from point source Guassian
  distribution root mean square displacement
  Xrms (2Dt)1/2

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First Passage Times

• How long does it take a molecules on average to
  travel a certain distance
• Diffusion limited rate
• Solve diffusion equation for geometry involved
  t x02/2D (1-D through distance x)
• Diffusion in the presence of an external force
• Kramers rate theory - Rate of reactions limited
  by diffusion over high-energy barrier

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Correlation Times

• How long will a molecule travel before thermal
  forces randomize the direction of motion
• Diffusion of a free protein