Mass, Stiffness, and Damping of Proteins

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Mass, Stiffness, and Damping of Proteins

• Chapter 3
• Howard

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Mass

• m ?V, where ? 1.38 x 103 kg/m3
• Mass in kg, molecular mass g/mole or dalton
  hydrogen atom mass of 1 dalton (1.66 x 10-24 g or
  1.66 x 10-27 kg)
• 100 kDa 166 x10-24 kg
• Volume 1.2 nm3 per kDa, 100 kDa 6 nm
  spherical diameter.
• 119.4 Da per amino acid, 7 amino acids per nm3

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Elasticity

• Homogenous mechanical properties identical
  throughout
• Isotropic properties do not depend on direction
• Strain ?L/L x Elastic (Youngs) modulus, E is
  proportional to pressure (F/A)
• E has same units as pressure (N/m2)
• E of some proteins similar to plexiglass
• Some proteins can be stretched up to 100 or more
  (yield pressure tensile strength)
• E does not depend on size or shape, but stiffness
  does. ? F/?L E(A/L)

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Protein Properties

• Material properties or atomic structures?
• Anisotropic - Domains connected by more flimsy
  connecting regions. Twisting is the result of
  connecting regions (reduced cross-section).
• Youngs moduli of several filamentous proteins
  similar. Direction of strain important hair.
• Molecular dynamics simulations

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Molecular Basis of Elasticity

• Stiffness of bonds covalent (strong),
  noncovalent (weak- van der waals, electrostatic)
• Bond energy depends on separation, r
• Energy profile parabolic
• Stretch a small distance (r-r0), with an applied
  force F(r) dU/dr??(r-r0), for large forces
  Hookes law breaks down. Max force (Fmax)
  reached and bond breaks

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Youngs Modulus

• Spring constant of each bond molecular dynamics
• Example Figure 3.3 a cubic lattice of springs
• F ??r, divide by r02 , F/r02 , force per unit
  area, r/r0 is the strain in each bond, E ?/r0
• High compliance (1/?)of proteins van der Waals
  bonds (weak links) dominate

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Model of Cubic Lattice of Springs
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Youngs Modulus of Proteins

• High compliance (1/?)of proteins van der Waals
  bonds (weak links) dominate
• Van de Waals potential energy - attactive
  dipole-dipole minus the steric repulsive force
• Expected Youngs modulus of filamentous proteins
  fits into this model 1 to 5 GPa
• Tensile strength 0.1 0.2 GPa
• Rigidity will be less if protein unfolded or
  unstable, variable conformations
• Rubber like elasticity deformation causes loss
  of entropy with alignment, makes unfavorable. As
  stretch they get stiffer, differs from rigid
  structures

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Viscous Damping

• Two damping forces viscosity of solution and
  interactions between amino acids
• Force per area (F/A) velocity gradient x the
  viscosity (?, Pa per sec)
• Newtonian fluid viscosity independent of
  velocity. Deviation shear thinning.
• Magnitude of drag force depends on Reynolds
  number(Re) ?Lv/? (ratio of inertial and viscous
  forces). Number less than 1, flow is laminar
  (creeping flow).
• Stokes law (Re lt1) Fd ??, ? 6??r

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Molecular Basis of Viscosity

• Associated drag force per molecule, ? p?ton,
  not dep. on speed if pton is independent of speed
• Drag depends on bonds, attachment time,
  stiffness of bonds
• Viscosity decreased with increased velocity, ton
  decreases (shear thinning)
• Force proportional to velocity gradient , wcton
  dv/dt
• Example viscosity of water
• Internal viscosity slows protein conformational
  changes does this limit conf. changes?

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Protein Friction Due to Cross-Links Between Two
Surfaces
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Global Motions of Proteins are Overdamped

• Viscous forces increase relative to inertial
  forces - Small objects expected to be overdamped
• Motion is overdamped if 4m?/?2 1
• Ratio scales with dimension, L
• Smaller object the smaller ratio, less tendency
  to oscillate
• How small must a protein be to ensure that its
  motion is overdamped?

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Why motions of large proteins are overdamped

• Rigidity of energy transducing proteins is lower
  than cytoskeletal proteins. Low value of
  stiffness leads to a greater characteristic
  length.
• Friction of the fluid like nature of the interior
  of proteins dampen out oscillations
• Elongated proteins are more highly damped than
  globular proteins. As aspect ratio increases, the
  damping increases while stiffness decreases.
• Protein filaments damping ratio increases as
  the length increases. Opposite to that of
  globular proteins

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Scale bar 50 nm
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Motions of Cytoskeleton and Cells are Overdamped

• Because the cytoskeleton is highly damped the
  cells are too
• High viscosity of cytoplasm still allows
  diffusion of smaller particles particles 50-500
  nm diameter are restricted
• Viscous forces still dominate even in very rigid
  highly organized cytoskeletal structure (cilia,
  flagella, stereocilia)
• Oscillations only occur on a macroscopic scale