Difference between revisions of "A Blind Spot in Confocal Reflection Microscopy: The Dependence of Fiber Brightness on Fiber Orientation in Imaging Biopolymer Networks"

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The authors do this for thousands of clusters which they then image using optical microscopy. For each value of N <math>\leq</math> 10 they determine different cluster configurations and their probabilities <math>P_i</math> and thus the free energies <math> F_i = -k_B T ln P_i </math>.
 
The authors do this for thousands of clusters which they then image using optical microscopy. For each value of N <math>\leq</math> 10 they determine different cluster configurations and their probabilities <math>P_i</math> and thus the free energies <math> F_i = -k_B T ln P_i </math>.
  
[[Image:jawerth1.jpg|400px|thumb|center|Fig. 1. A. Experimental system: microwells filled with polyNIPAM microgel particles and PS microbeads. B. Close-up view of two sticky microspheres and surrounding nanoparticles. C. Pair potential. Note that the depletion attraction is very short-range, so the interaction is strictly pairwise additive. D. Micrograph of several microwells, each with different individual clusters. E. Optical micrographs of colloidal clusters in microwells with N = 2, 3, 4, 5 particles.]]
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[[Image:jawerth1.jpg|400px|thumb|center|Fig. 1. Simultaneous imaging of a collagen network using CRM and CFM. A. Typical image from CRM. B. Corresponding image using CFM. Red circles highlight fibers that do not appear in the reflection image. C. Overlay of panels A (green) and B (red). D. Projection of 50 x,z slices along the y axis using CRM image data. E. Equivalent projection using CFM. F. Overlay of panels D (green) and E (red). ]]
  
 
== Results ==
 
== Results ==
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Cases of N = 7 and N = 8 are similar to N = 6, although the number of structures increases to 6 and 16. For all cases, all the structures have the same number of contacts and entropic effects dominate, with highly symmetric structures extremely unfavored. These are all shown in Fig 2.  
 
Cases of N = 7 and N = 8 are similar to N = 6, although the number of structures increases to 6 and 16. For all cases, all the structures have the same number of contacts and entropic effects dominate, with highly symmetric structures extremely unfavored. These are all shown in Fig 2.  
  
[[Image:jawerth2.gif|400px|thumb|center|Fig. 2. Comparison of experimental and theoretical cluster probabilities P at N = 6, 7, and 8. Structures that are difficult to differentiate experimentally have been binned together at N = 7 and 8 to compare to theory. The calculated probabilities for the individual states are shown as light gray bars, and binned probabilities are dark gray. Orange dots indicate the experimental measurements, with 95% confidence intervals given by the error bars.]]
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[[Image:jawerth2.gif|400px|thumb|center|Fig. 2. Relative frequency of the moment angle <math>\theta</math> for CFM data (triangles) and CRM data (circles) in both rotated (solid) and non-rotated (open) samples. Light-shaded line: sine distribution expected for an isotropic sample. Inset: corresponding <math>\phi</math> distributions.]]
  
 
The landscape undergoes a qualitative change when N reaches 9. The number of structures predicted theoretically reaches 77 for N = 9 and 393 for N = 10, too many to catalog experimentally. The authors concentrate on measuring the probability of having structures of two types:  
 
The landscape undergoes a qualitative change when N reaches 9. The number of structures predicted theoretically reaches 77 for N = 9 and 393 for N = 10, too many to catalog experimentally. The authors concentrate on measuring the probability of having structures of two types:  
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2. structures with more than 3N - 6 bonds. These often have high symmetry, but they have extra bonds. The potential-energy gain is therefor large enough to overcome the deficiency in rotational entropy. (Fig 3C).  
 
2. structures with more than 3N - 6 bonds. These often have high symmetry, but they have extra bonds. The potential-energy gain is therefor large enough to overcome the deficiency in rotational entropy. (Fig 3C).  
  
[[Image:jawerth3.gif|400px|thumb|center|Fig. 3. A. Optical micrographs and renderings of nonrigid structures at N = 9 and (B) N = 10 (C) Structures of 3N – 5 = 25 bond packings at N = 10. The anharmonic vibrational modes of the nonrigid structures are shown by red arrows. Experimentally measured probabilities are listed at top. Annotations in micrographs indicate clusters corresponding to subsets of HCP or FCC lattices. Scale bars, 1 µm.]]
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[[Image:jawerth3.gif|400px|thumb|center|Fig. 3. Intensity of individual fibers as a function of their <math>\theta</math>-angle for CFM (triangles) and CRM (circles) for both the rotated (open) and nonrotated (solid) cases. Shaded line shows expected values from theory. ]]
  
 
While the average probabilities at N = 9 and N = 10 should be 1% and 0.25%, the single nonrigid structure at N = 9 has probability 10%, and the 3 extra-bond structures at N = 10 occur 10% of the time. This confirms that nonrigid structures and structures with extra bonds are favored.
 
While the average probabilities at N = 9 and N = 10 should be 1% and 0.25%, the single nonrigid structure at N = 9 has probability 10%, and the 3 extra-bond structures at N = 10 occur 10% of the time. This confirms that nonrigid structures and structures with extra bonds are favored.

Revision as of 17:16, 24 October 2010

Entry by Leon Furchtgott, APP 225 Fall 2010.

A Blind Spot in Confocal Reflection Microscopy: The Dependence of Fiber Brightness on Fiber Orientation in Imaging Biopolymer Networks. L.M. Jawerth, S. Munster, D.A. Vader, B. Fabry, and D.A. Weitz. (2010). Biophysical Journal, 98, L01-L03.

Summary

The paper is interested in imaging techniques for biopolymers. In particular, the paper compares the effectiveness in imaging collagen networks of two confocal imaging methods, confocal reflection microscopy (CRM) and confocal fluorescence microscopy (CFM). The authors simultaneously image collagen using the two techniques and find that in CRM, fiber brightness depends strongly on fiber orientation. This explains why when using CRM to image collagen, the network appears to be aligned with the imaging plane, whereas in CFM, the network seems isotropic.

Background

Difference between CRM and CFM:

Previous research in collagen network architecture:

Experimental Setup

Small numbers of polystyrene (PS) microspheres were placed in cylindrical microwells filled with polyNIPAM nanoparticles (Fig 1A, 1D). The microwells have depth and diameter 30 <math>\mu</math>m, and they are chemically functionalized so that the particles cannot stick to the surfaces. The microspheres have diameter 1.0 <math>\mu</math>m. The nanoparticles have diameter 80 nm and induce a depletion attraction between 2 microspheres (sticky spheres, see Fig 1B). This depletion attraction is very short-ranged (< 1/10 PS sphere diameter) which means that the interactions are pairwise additive (see Fig 1C). Therefore the total potential energy U of a given structure is well approximated by <math>U = CU_m</math>, where <math>C</math> is the number of contacts or depletion bonds and <math>U_m</math> is the depth of the pair potential.

The authors do this for thousands of clusters which they then image using optical microscopy. For each value of N <math>\leq</math> 10 they determine different cluster configurations and their probabilities <math>P_i</math> and thus the free energies <math> F_i = -k_B T ln P_i </math>.

Fig. 1. Simultaneous imaging of a collagen network using CRM and CFM. A. Typical image from CRM. B. Corresponding image using CFM. Red circles highlight fibers that do not appear in the reflection image. C. Overlay of panels A (green) and B (red). D. Projection of 50 x,z slices along the y axis using CRM image data. E. Equivalent projection using CFM. F. Overlay of panels D (green) and E (red).

Results

The cluster classifications can then be compared to experimental predictions, which are found in a previous theoretical PRL paper (Arkus, Manoharan, Brenner. Phys. Rev. Lett. 103, 118303 (2009)). Observed structures agree with experimental predictions. For N = 2, 3, 4, 5, one unique structure (dimer, trimer, tetrahedron, triangular dipyramid), as shown in Fig 1E.

The first interesting case is N = 6. Two structures are observed, an octogon and a polytetrahedron. Both have C = 12 contacts and thus have the same potential energy. What explains the 20-fold preference for the polytetrahedron (3 kT difference) is an entropic difference. The entropy can be divided into two factors: a rotational entropy and a vibrational entropy. The rotational entropy makes the largest contribution to the free-energy difference and is proportional to the moment of inertia or equivalently the permutational degeneracy. This contributes a factor of 12, the remaining factor of 2 coming from the vibrational entropy. In the case of N = 6 as for N = 7 and 8, highly symmetric structures are extremely unfavorable among clusters with the same potential energy.

Cases of N = 7 and N = 8 are similar to N = 6, although the number of structures increases to 6 and 16. For all cases, all the structures have the same number of contacts and entropic effects dominate, with highly symmetric structures extremely unfavored. These are all shown in Fig 2.

Fig. 2. Relative frequency of the moment angle <math>\theta</math> for CFM data (triangles) and CRM data (circles) in both rotated (solid) and non-rotated (open) samples. Light-shaded line: sine distribution expected for an isotropic sample. Inset: corresponding <math>\phi</math> distributions.

The landscape undergoes a qualitative change when N reaches 9. The number of structures predicted theoretically reaches 77 for N = 9 and 393 for N = 10, too many to catalog experimentally. The authors concentrate on measuring the probability of having structures of two types:

1. nonrigid structures, in which one of the vibrational modes is a large-amplitude, anharmonic shear mode. These have high vibrational entropy. (Fig 3A, 3B).

2. structures with more than 3N - 6 bonds. These often have high symmetry, but they have extra bonds. The potential-energy gain is therefor large enough to overcome the deficiency in rotational entropy. (Fig 3C).

Fig. 3. Intensity of individual fibers as a function of their <math>\theta</math>-angle for CFM (triangles) and CRM (circles) for both the rotated (open) and nonrotated (solid) cases. Shaded line shows expected values from theory.

While the average probabilities at N = 9 and N = 10 should be 1% and 0.25%, the single nonrigid structure at N = 9 has probability 10%, and the 3 extra-bond structures at N = 10 occur 10% of the time. This confirms that nonrigid structures and structures with extra bonds are favored.

The full theoretical free-energy landscape is shown in Fig 4.

Discussion

The paper shows/confirms that the most stable small clusters of hard spheres with short-ranged attractions can be determined by geometrical rules: (1) rotational entropy favors structures with fewer symmetry elements; (2) vibrational entropy favors nonrigid clusters, which have half-octahedral substructures sharing at least one vertex; and (3) potential energy favors clusters with both octahedral and tetrahedral substructures, allowing them to have extra bonds.

The paper's description of free-energy landscape is still incomplete because of some of the simplifications it makes. In particular, while the interaction energy in the paper is extremely short-ranged, this is not generally true. For longer-range interactions, the effects will no longer be fundamentally entropic, and they will depend on temperature as well.

Relation to Soft Matter

This is a neat paper that fits well with the first few lectures and readings in our course. It makes you think about fundamental ideas in statistical mechanics and how to measure them experimentally. It also shows how complex configurations can quickly come about from simple short-range interactions.