Statistical methods for assessing the dimensions of synaptic vesicles in nerve terminals

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Abstract

Chemical transmission between neurons occurs by the release of neurotransmitter packaged within vesicles of the presynaptic neuron onto a postsynaptic target. The amount of transmitter contained within a vesicle is in part regulated by the size of the vesicle. Thus, it is of general interest to quantify the dimension of vesicles in understanding the basic principles of chemical synaptic transmission. These vesicles can only be measured by electron microscopic techniques. Obtaining the true dimensions of synaptic structures is therefore complicated by stereological considerations. In this study, we suggest improved methods for determining the distributions (and mean sizes) for populations of vesicle diameters by mathematical processes involving (1) an implicit inversion of the empirical data distribution, (2) an explicit inversion approach, and (3) an approach based on substituting the empirical distribution into the inversion formula and then isotonizing using an iterated convex minorant algorithm. These procedures provide distributions that better represent the true population distributions (and means) for comparisons with other data sets of vesicle diameter measures.

Introduction

Characterizing structural entities of synapses, which mediate communication between nerve cells and their targets is important for defining the limits of cellular interactions. Synaptic vesicles within nerve terminals package and store neurotransmitter substances and when released during exocytosis, cause a reaction in the receiving (postsynaptic) cells. Synaptic vesicles are concentrated within the presynaptic nerve terminal and release their contents when there is an intracellular rise of calcium ions (Ca2+) in the nerve terminal due to electrical depolarization. The amount of neurotransmitter stored and released by a vesicle is one of the factors that influences the size of the postsynaptic response (Zhang et al., 1998). Thus, changes in the size of synaptic vesicles through genetic or physiological alterations have important implications for neuronal signaling. For example, stimulation in the hippocampus of rats to induce long-term potentiation induced depletion of transmitter and a reduction in the observable vesicle diameters within the presynaptic terminals. The reduction in vesicle size was reversible when physiological recovery resumed (Petukhov and Popov, 1986). Also, in Drosophila, recent studies of mutants have shown that quantal size is related to vesicle size. Examples include the shibire mutation, in which the vesicles become increased in size (van de Goor et al., 1995, Zhang et al., 1998). Increasingly, genetic mutations which effect synaptic function are being studied in Drosophila, and the relationship between synaptic structure and function requires precise assessment of synaptic structures and matching dimensions of synaptic vesicles (Atwood and Cooper, 1995, Cooper et al., 1995a, Cooper et al., 1995b, Cooper et al., 1996, Propst and Ko, 1987, Wong et al., 1999).

In order to assess such structure–function relationships, careful measure of synaptic vesicles and the limitations of this measurement need to be addressed. However, measurement of vesicular dimensions is particularly problematic because of their small size; they can only be visualized for measurement in micrographs obtained by electron microscopy. Specimens of nerve terminals optimal for conventional electron microscopy are obtained when the tissue is sectioned in the range of 50–100 nm in thickness. Vesicles are sectioned in a random orientation; some vesicles are transected at various planes, while others reside within the section and between the sectioning planes, depending on the section thickness and vesicle dimensions. Determination of the spherical diameters offers stereological problems that need to be addressed in order to characterize the dimensions of the vesicles and the population of diameter measurements for comparative purposes. The measurement problem of spherical vesicles occurs when vesicles are sectioned at the caps, producing various sized projected circular images. When the center of the spherical vesicle resides within the section, the true diameter of the sphere will be observed in the projected image. The projection of the varying sized caps (when the center of the spherical vesicle lies outside the section) and the complete vesicles from 3-D space on to the 2-D viewing plane will result in observed circles of various diameters. Theoretically, the smallest of the vesicle caps should also be observable, but in practice this is not the case. So the observable data is degraded in at least two ways and does not represent the true distribution of the randomly sectioned vesicles.

Most investigators report the mean diameter of measured vesicles from electron micrographs, but if the distribution is made up of multiple sized vesicles with varying amounts in the sample, a ‘mean’ value represents a mixed population and includes the diameters of the ‘caps’ (degraded data). In order to better represent a mean for such distributions we make use of a mathematical approach that corrects for the biases in the diameters of the distribution. This study illustrates that obtaining a mean value from the measurable diameters within a combined distribution does not yield the true diameter of the actual vesicles in the sample, since fragments of the vesicles distort the distribution and because populations of vesicles with different mean sizes may be present. By correcting the distortion of the observed distribution of measurable vesicle diameters, a closer approximation of the true distribution is achieved.

Section snippets

Methods

We used electron micrographs of synapses obtained from the abdominal slow flexor muscle in the crayfish, Procambarus clarkii, supplied by Atchafalaya Biological Supply Co. (Raceland, LA). Preparations were dissected in a modified Van Harreveld’s crayfish solution (Wojtowicz and Atwood, 1984). Procedures for processing for electron microscopy are described by Jahromi and Atwood (1974). Sections were collected on Formvar-coated slotted grids. A log of each serial section was kept and the

Results

The mathematical background for the stereological problems considered here are based on the assumption that spheres (neglecting the possibility of overlapping) are homogeneously distributed in three-dimensional space, and that observations are taken within a given random ‘slice’ through this space, defined by means of two parallel planes of distance 2μ apart. We let f(·) denote the true density function for the radii of the spheres in three-dimensional space, and g(·) denote the density

Discussion

The amount of transmitter contained within a vesicle is one of the key factors which determines the degree of transmission from one cell to another. The amount of neurotransmitter in a vesicle is believed to be related to the size of the vesicle (Zhang et al., 1998). Due to the small sizes of synaptic vesicles (40–60 nm diameters) at the crayfish NMJ, electron microscopic techniques provide the only approach to image the vesicle dimensions. Obtaining the true dimensions of synaptic structures

Acknowledgements

We thank Leo Marin for tissue processing and photomicrographs of tissue, and Brenda Crowe for programming assistance. Funded by NSERC-Canada (A. Feuerverger), NSF grant IBN-9808631 (R.L. Cooper) and MRC-Canada (H.L. Atwood).

References (41)

  • A.J Baddeley et al.

    Estimation of surface area from vertical sections

    J. Microsc.

    (1986)
  • R.A Becker et al.

    The New S Language

    (1988)
  • K.R Clark et al.

    Unbiased linear property estimation for spheres, from sections exhibiting over projection and truncation

    J. Microsc.

    (1983)
  • R Coleman

    An Introduction to Mathematical Stereology

    (1979)
  • R.L Cooper et al.

    Measurement problems associated with the reconstruction of synaptic structures at the electron microscopic level

    Abstr. Soc. Neurosci.

    (1995)
  • R.L Cooper et al.

    Synaptic differentiation of a single motor neuron: conjoint definition of transmitter release, presynaptic calcium signals, and ultrastructure

    J. Neurosci.

    (1995)
  • R.L Cooper et al.

    Synaptic structural complexity as a factor enhancing probability of calcium-mediated transmitter release

    J. Neurophysiol.

    (1996)
  • L.M Cruz-Orive

    On the estimation of particle number

    J. Microsc.

    (1980)
  • L.M Cruz-Orive et al.

    Recent stereological methods for cell biology: a brief survey

    Am. J. Physiol.

    (1990)
  • D.M.G DeGroot et al.

    The complex-shaped ‘perforated’ synapse, a problem in quantitative stereology of the brain

    J. Microsc.

    (1982)
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