Chapter 20: Coulombs Law of Electrostatic Forces

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Chapter 20Coulombs Law of Electrostatic Forces

• BME 531
• Nathan Baker
• baker_at_biochem.wustl.edu

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Why care about electrostatics?

• Longest-range interactions
• Cannot be accurately truncated
• Diverge at origin and infinity
• Important for all charged particles
• Solvated ions
• Biomolecules
• Plasmas
• Ionic liquids
• Defects in solids

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Biomolecular charge distributions proteins

• Amphoteric range of titratable groups
• Range of isoelectric points (calculated)
• End result zwitterionic with a wide range of
  charge densities

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Biomolecular charge distributions nucleic acids

• dsDNA
• Approx. linear form
• Close phosphate spacing
• B-form phosphate spacing 3.4 Å
• RNA
• Structural diversity
• Dense phosphate packing

11 bp B-form DNA (1AGH)
23s rRNA (1FFZ)
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Biomolecular charge distributions other
molecules

• Sugars
• Glycosaminoglycans (GAGs)
• Cellular matrix
• Cell surface co-receptors
• Potentially high charge density
• Proteoglycans GAGs attached to proteins by
  glycosylation
• Membranes
• Phospholipids
• Zwitterionic (phosphatidyl-choline,
  phosphatidyl-ethanolamine, sphingo-myelin)
• Anionic (phosphatidyl-serine)
• Other components

Dermatan sulfate. (Picture from Alberts et al)
Electrostatic potential of POPC membrane.
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History of electrostatics

• Basic principles established for macroscopic
  objects in late 1700s
• Analysis of interactions between charged objects
• Phenomenological model
• Applicable over 25 orders of magnitude in length
• Earths magnetic field (107 m)
• Coulomb experiments (100 m)
• a particle scattering (Rutherford, 10-13 m)
• Electron-positron scattering (QED, 10-18 m)

Schematic of Cavendish apparatus used by Coulomb.
Picture from http//www.fas.harvard.edu/scdiroff
/lds/NewtonianMechanics/CavendishExperiment/Cavend
ishExperiment.html
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Coulombs law

• Every model uses Coulombs law (somewhere)
• Phenomenological model circa 1785 for
  charge-charge interactions in a vacuum
• Relates potential to charge for homogeneous
  dielectric materials
• Provides superposition of potentials
• Assumptions
• Vacuum
• Point charges
• No mobile ions
• Infinite boundaries

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Electrostatics uses a bewildering number of unit
conventions

• Please use SI units
• Basic unit system can be identified by looking
  for the 4pe0

Charge (C)
Energy (J)
Vacuum permittivity (8.85410-12 C2 J-1 m-1)
Distance (m)
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SI units

• Charge C
• Energy J
• Distance m
• Potential V J C-1
• Capacitance F C V-1

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Charge interactions are long-ranged
r-1, r-6, r-12, e-r

• Decays much more slowly than other interactions
• The Coulomb potential cannot be integrated over
  an infinite domain
• Sums related to electrostatic interactions (see
  next example) are conditionally convergent

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Electrostatic interactions in NaCl crystals

• Assume ions are in rigid lattice with spacing a
  2.81 Å
• Sum Na interactions over the first row
• Sum Na interactions over the 4 adjacent rows
• Energy not decreasing very rapidly!

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Electrostatic interactions in NaCl crystals

• Continue this process to the next-nearest
  neighbors, etc.
• Result Madelung constant
• Use Madelung constant to calculate lattice energy
• Underestimate of 96 kJ mol-1 due to several
  factors, including
• Assumption of incompressibility
• Assumption of fixed lattice spacing

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Electrostatic interactions are strong

• Electrostatic interactions are much stronger than
  most other non-bonded interactions e.g.,
  gravitational
• Charge imbalance has serious energetic penalties
• Lightning
• Static electric shock 1000 charges
• Solution charge imbalance
• 1000 charges
• 1 part in 1021
• Electroneutrality of a macroscopic solution is a
  reasonable assumption

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Electrostatics in dielectric media

• A continuum dielectric medium
• Has no atomic detail
• Is related to polarization of the medium
  redistribution of charges
• Responds linearly and locally to dampen an
  applied field
• Is characterized by a dielectric tensor
• Reduces the strength of electrostatic
  interactions relative to a vacuum

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Electrostatics in dielectric media

• An isotropic dielectric continuum exhibits the
  same response in all directions
• The dielectric tensor can be reduced to a scalar
• For a homogeneous isotropic Coulombs law takes a
  very simple, scaled form

Dielectric coefficient (unitless)
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Dielectric coefficients

• Several contributions to polarizability
• Reorientation of permanent dipole moment
• Molecular electronic and nuclear polarizability
• Hydrogen bonding networks

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Bjerrum length fundamental length scale for
Coulombs law

• What is the distance at which two unit charges
  interact with kT of energy? Bjerrum length
• A useful length scale for determining when
  electrostatic interactions are on the same order
  as thermal energy
• Provides simpler form for Coulombs law
• Approximately 7 Å for water (D 80) at 298 K

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Electrostatic energy of NaCl in water

• Same situation as before
• Interionic distance 2.81 Å
• 1 e charges
• Bjerrum length changes dramatically in water
• Ion pair interaction energy decreases
• Crystal is significantly less stable salt
  dissociates in water but not in lower dielectrics

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Electrostatic forces

• Force is the negative gradient of the potential
• Assume all other terms are constant (homogeneous
  medium)
• Force is vector-valued

Force on charge B due to charge A.
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Electrostatic superposition

• For a homogeneous system
• Total electrostatic potential is the sum of
  individual electrostatic potentials
• Total electrostatic force is the sum of
  individual electrostatic forces
• This works for arbitrary charge distributions
• This is because Coulombs law is a Green
  function for a particular partial differential
  equation (coming up)

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Electrostatic fields and potentials

• Potential
• What is the energy of placing a unit charge at
  position x?
• A scalar-valued function
• Factoring charge (C) out of energy (J) gives
  units of V J C-1
• Field
• What is the force experienced by a unit charge at
  position x?
• A vector-valued function
• Factoring charge (C) out of force (N J m-1)
  gives units of N C-1
• Superposition applies potentials and forces can
  be added
• Purpose a good way to represent the
  electrostatics of a charge distribution

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Electrostatic fields
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Electric field flux

• Flux the amount of stuff passing through
  surface
• Concentration
• Fluid flow
• Electric field
• Fluxes arise from
• Sources positive charges
• Sinks negative charges
• Electric field flux integral of electric
  displacement over a surface

Jacobian points in surface normal direction
Electric displacement
Boundary surface of volume O
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Field flux point charge in a sphere

• Point charge has spherically-symmetric field
• Field is constant on sphere surface
• Flux is independent of sphere diameter

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Field flux point charge in a balloon

• Consider another outer surface that surrounds
  an inner sphere
• The outer surface can have any shape
• The fluxes through any (arbitrarily small)
  portion of the outer and inner surfaces can be
  calculated
• These surface portions can be related
• The fluxes through the two surfaces are the same!

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Gauss law

• The integral of field flux through a closed,
  simple surface is equal to the total charge
  inside the surface
• This is true for both homogeneous and
  inhomogeneous dielectric media
• This generalizes to other charge distributions

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Field from a line charge

• Suppose we have
• Homogeneous medium
• Line of length L, where L is very big (radial
  symmetry)
• Linear charge density of ?
• What is the field at distance r from the source?
• Compute the flux through a cylindrical surface
• Calculate the enclosed charge
• Use Gauss Law

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Field around DNA

• Assume B-DNA shape
• 2 phosphates every 3.4 Å
• Water dielectric of 80
• What is the field 40 Å away from a very long
  B-DNA molecule?

Image from Stryer Biochemistry
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Field from a charged plane

• Suppose we have
• Homogeneous medium
• Surface of area A, where A is very big (one
  dimensional)
• Surface charge density of s
• What is the field at distance r from the source?
• Compute the flux through a pillbox
• Calculate the enclosed charge
• Use Gauss Law

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Field around a membrane

• POPS membrane
• -1 e charge per lipid
• 1 lipid per 55 Å2
• What is the field 20 Å away from the membrane (in
  water)?
• Bigger than DNA!

Mukhopadhyay P, et al. Biophys J 86 (3) 1601-9,
2004.