Measuring the Length Distribution of a Fibril System:

1
Measuring the Length Distribution of a Fibril
System a flow birefringence technique
applied to amyloid fibrils
Salman S Rogers,1 Paul Venema,2 Leonard Sagis,2
Erik van der Linden,2 and Athene M Donald1 1 BSS
Sector, Cavendish Laboratory, Cambridge
University, Cambridge CB3 0HE, UK. 2
Laboratory of Food Physics, Wageningen
University, PO Box 8129, 6700EV Wageningen, The
Netherlands. Correspondence e-mail
salman.rogers_at_physics.org

• 1. Aim to measure the length distribution of a
  solution of b-lactoglobulin fibrils, which are
• very polydisperse in length 1mm
• semiflexible
• in semidilute solution (i.e. entangled)

• 6. Comparison with TEM
• The fitted length distribution has two missing
  parameters
• M optical anisotropy per unit length
  concentration of the fibrils,
• b rotational diffusivity prefactor,
• which can be determined with other experiments
  we determined the fractional conversion of
  monomers to fibrils with sedimentation, and
  compared the fibril lengths with TEM
  measurements, to find M1.74?10-20 and b6.5?104.
• Fig 3. Rheo-optics and TEM distributions match
  with M, b as above

Abstract Relaxation of flow birefringence can
give a direct measure of the rotational diffusion
of rod-like objects in solution. With a suitable
model of the rotational diffusivity, a length
distribution can be sought by fitting the decay
curve. We have measured the flow birefringence
decay from solutions of amyloid fibrils composed
of b-lactoglobulin, and extracted a length
distribution using the Doi-Edwards-Marrucci-Grizzu
ti theory of semidilute rotational diffusion. The
concentration scaling of the results show that
the fibrils diffuse as free rods they cannot be
significantly branched, sticky or break up under
dilution. The length distribution obtained shows
a single broad peak, consistent with
measurements of the fibrils by electron
microscopy. This comparison, and combination of
the experiment with an assay to find the total
concentration of fibrils, allows calibration of
the length scale and concentration scale of the
length distribution. It is our hope that this
method can be used for following the growth
kinetics of amyloid fibrils in vitro, and for
studying the length distribution of rod-like
systems in general.

• 2. Motivation
• The fibrils, which self-assemble from
  b-lactoglobulin monomers, are examples of amyloid
  fibrils. Other amyloid fibrils are involved in
  various degenerative diseases, and an
  understanding of the assembly kinetics of the
  fibrils would be important in the study of these
  diseases. A versatile technique of measuring the
  length distribution would provide a powerful
  experimental approach to the assembly kinetics.
• There is interest in using b-lactoglobulin
  fibrils in food or biomaterials. The mechanical
  properties of such materials would depend
  crucially on the length of the fibrils.
• There are no easy methods of measuring a length
  distribution of fibrils like these.

• 4. Inverting the data
• The scaled decay curve can be fitted
• using the DEMG model.
• First we estimate the length distribution
• using a linearly regularised inverse
• Laplace transform.
• Then we iteratively adjust the distribution until
  it fits the measured birefringence decay.
• Fig 2. Fitting the decay with various initial
  estimates shows fitting error due to noise is
  modest. (a) measured/fitted decays (b)
  length dist.

• 7. Conclusion
• We have developed a promising technique for
  measuring a fibril length distribution, which can
  potentially be applied to any fibril system whose
  orientational relaxation can be resolved in time.
  The errors in the distribution can be evaluated
  quantitatively.
• Coming soon
• electric birefringence measure of the short end
  of the distribution,
• b-lactoglobulin length distributions in
  different solution conditions.

• 3. Approaching the problem rheo-optics
• In shear flow, the fibrils align, leading to
  measurable flow birefringence. When the flow is
  stopped, the birefringence decays on a spectrum
  of time scales, as the fibrils diffuse
  rotationally. Short fibrils diffuse faster than
  longer ones, so the decay curve contains
  information about the range of lengths in the
  system.
• The decay curve can be fitted quantitatively
  using a theory of diffusivity of rods we use the
  Doi-Edwards-Marrucci-Gruzzuti (DEMG) model, which
  describes a polydisperse, semidilute solution.
• Fig 1. Scaling of decay time with (conc.)2 shows
  the applicability of DEMG

Fig 4. Rheo-optics length distribution with
errors from fitting, fibril stretching, sample
inertia and incomplete alignment considered
Now published Macromolecules 2005, 38(7)
29482958 DOI 10.1021/ma0474224 on web since
3 March 2005
(Background image b-lactoglobulin fibrils imaged
by TEM, coloured digitally)