# Nonequilibrium Fluctuations in Metaphase Spindles: Polarized Light Microscopy, Image Registration, and Correlation Functions

## Contents

## Basic Information

Wiki by Bryan Kaye

**Title:** Nonequilibrium Fluctuations in Metaphase Spindles: Polarized Light Microscopy, Image Registration, and Correlation Functions

**Authors:** Jan Brugu ́es and Daniel J. Needleman

**Article URL:** http://www.needleman.seas.harvard.edu/papers/dan/Proc%20SPIE%20Feb%202010.pdf

**Keywords:** Mitotic Spindle, Polarized Light Microscopy, Correlation Function, Image Registration, nematic phase, order parameter and optical axis

First I'll define some things you'll need to understand the concepts in the paper and therefore my wiki. I had to look up most this stuff myself, so there is nothing to feel bad about if this is new to you. I looked up this stuff in wikipedia, so some of it directly quoted from wikipedia, some paraphrased, and some my own.

## Summary

This paper studies the internal fluctuations of spindles by computing spatio-temporal correlation functions. The correlation function of a specific point on the spindle can only be calculated by accounting for the net motion of the spindle. The paper puts forth a method for accounting for spindle motion (image registration) and puts estimates on its error. They then do a bunch of math that shows how you can calculate nematic orders from correlation functions. Therefore studying these correlation functions can help test coarse-grained theories of the spindle.

## Definitions

**Non-Equilibrium Steady State:** There is entropy production and some flows are non-zero, but there is no time variation

**Nematic Order:** In a nematic phase, the calamitic or rod-shaped organic molecules have no positional order, but they self-align to have long-range directional order with their long axes roughly parallel. Thus, the molecules are free to flow and their center of mass positions are randomly distributed as in a liquid, but still maintain their long-range directional order. Most nematics are uniaxial: they have one axis that is longer and preferred, with the other two being equivalent (can be approximated as cylinders or rods). Therefore microtubules have a nematic order.

**Coarse-Grained Model:** A fine-grained description of a system is a detailed, low-level model of it. A coarse-grained description is a model where some of this fine detail has been smoothed over or averaged out. The replacement of a fine-grained description with a lower-resolution coarse-grained model is called coarse graining. (See for example the second law of thermodynamics)

**Order parameter** :The local nematic director, which is also the local optical axis, is given by the spatial and temporal average of the long molecular axes
The description of liquid crystals involves an analysis of order. A tensor order parameter is used to describe the orientational order of a liquid crystal, although a scalar order parameter is usually sufficient to describe nematic liquid crystals. To make this quantitative, an orientational order parameter is usually defined based on the average of the second Legendre polynomial.

<math> S = <\frac{3*cos^{2}(\theta)-1}{2}> </math>

where θ is the angle between the LC molecular axis and the local director (which is the 'preferred direction' in a volume element of a liquid crystal sample, also representing its local optical axis). The brackets denote both a temporal and spatial average. This definition is convenient, since for a completely random and isotropic sample, S=0, whereas for a perfectly aligned sample S=1. For a typical liquid crystal sample, S is on the order of 0.3 to 0.8, and generally decreases as the temperature is raised. In particular, a sharp drop of the order parameter to 0 is observed when the system undergoes a phase transition from an LC phase into the isotropic phase. The order parameter can be measured experimentally in a number of ways. In our case, we use birefringence to measure the order parameter.

**Optical Axis:** This is NOT the axis that light travels down through the microscope. Instead, this is the axis that the microtubules are aligned with. Therefore the optical axis can be different for point in the spindle.

**Birefringence:** Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials. The simplest instance of the effect arises in materials with uniaxial anisotropy. That is, the structure of the material is such that it has an axis of symmetry with no equivalent axis in the plane perpendicular to it. (Cubic crystals are thereby ruled out.) This axis is known as the optical axis of the material, and light with linear polarizations parallel and perpendicular to it has unequal indices of refraction, denoted ne and no, respectively, where the suffixes stand for extraordinary and ordinary. The names reflect the fact that, if unpolarized light enters the material at a nonzero acute angle to the optical axis, the component with polarization perpendicular to this axis will be refracted as per the standard law of refraction, while the complementary polarization component will refract at a nonstandard angle determined by the angle of entry and the difference between the indices of refraction, known as the birefringence magnitude.

**Xenopus Laevis:** is a species of South African aquatic frog of the genus Xenopus. This frog is commonly used in spindle research because it has a large and easily manipulable embryo.

## Introduction

Many of the molecules that make up the spindle are known, but is not clear how these components work together to make a working spindle. Microscopic models of the spindle seem very difficult to do and very far away. However, phenomenological models, like those from liquid crystals, can be applied to the spindle. In order to test these models, one has to compare experiment with theory. , Needleman lab looks at the fluctuations (aka correlation function) of the birefringence in order to test coarse-grained models of the spindle.

However, in order to compute the correlation function, you need to take many pictures of the same spot of the spindle. This is difficult because the spindle as a whole moves with respect to the microscope. Needleman thinks this is due to convective flows in the chamber and protein motors from the spindle on the cover slip.

## Numerical Methods

There are two ways Needleman aligns the spindle. The first method is based on identifying key features (shapes) of the spindle and translating/rotating the image until the shapes match after each frame. While this method is simple and intuitive, it is not very robust because the spindle has slight shape changes over time.

The second method uses a bulk correlation (looking over the ENTIRE spindle) method to align frames. This method first blocks out the background (non-spindle) so that objects passing in the background (vesicles) do not introduce misalignments. Then it tries translating and rotating the image such that the correlation of every pixel in the current image with previous image is a maximum.

Furthermore, "The images are only aligned after the entire algorithm has completed. Updating the alignment for each frame directly after determining its rotation and displacement magnitude would inevitably increase the registration error, since rotation and (subpixel) translation of the spindle involves interpolation which would propagate to subsequent frames. Therefore the registration algorithm only uses the retardance images, which have a higher dynamical range and more pronounced internal features than the orientation images."

## Error Estimation of Alignment

We want to estimate the error in the alignment procedure. One way to do this is to register an image with itself after adding gaussian noise to the image. The alignment procedure is very robust against noise and, using estimates of gaussian noise of spindle, should give only a ~0.2 degree error in rotation and ~0.1 pixel error in translation. However, this estimate only takes into account error due to gaussian noise, which is not the sole cause of the real variation between consecutive images.

## Physical Methods

Needleman uses Xenopus Laevis oocyte (egg) spindles. Pictures are taken by with a polarized light microscope, called an LC-Polscope. For each pixel in an image, the LC-Polscope measures the retardance - the sample birefringence integrated over the optical volume - and the orientation of the optical slow axis. The measured retardance is determined by the number of microtubules in an optical volume and their degree of alignment.

## Results and Conclusion

Using the alignment tools described above, Needleman shows how one can calculate microtubule orientation and estimate the correlation function of the nematic order parameter of the microtubules.

I think this paper is really more about the alignment procedure than anything else. In addition, this paper lies the groundwork for calculating quantities that can be used to test coarse-grained models of the spindle. However, no specific models are tested in this paper. For me, the hardest thing about this paper was becoming familiar with all the terms and theories.