Robust ab initio solution of the cryo-EM reconstruction problem at low resolution with small data sets

Single particle cryo-electron microscopy has become a critical tool in structural biology over the last decade, able to achieve atomic scale resolution in three dimensional models from hundreds of thousands of (noisy) two-dimensional projection views of particles frozen at unknown orientations. This is accomplished by using a suite of software tools to (i) identify particles in large micrographs, (ii) obtain low-resolution reconstructions, (iii) refine those low-resolution structures, and (iv) finally match the obtained electron scattering density to the constituent atoms that make up the macromolecule or macromolecular complex of interest. Here, we focus on the second stage of the reconstruction pipeline: obtaining a low resolution model from picked particle images. Our goal is to create an algorithm that is capable of ab initio reconstruction from small data sets (on the order of a few thousand selected particles). More precisely, we seek an algorithm that is robust, automatic, able to assess particle quality, and fast enough that it can potentially be used to assist in the assessment of the data being generated while the microscopy experiment is still underway.Comment: 39 pages, 12 figure

Functional roles for synaptic-depression within a model of the fly antennal lobe.

Several experiments indicate that there exists substantial synaptic-depression at the synapses between olfactory receptor neurons (ORNs) and neurons within the drosophila antenna lobe (AL). This synaptic-depression may be partly caused by vesicle-depletion, and partly caused by presynaptic-inhibition due to the activity of inhibitory local neurons within the AL. While it has been proposed that this synaptic-depression contributes to the nonlinear relationship between ORN and projection neuron (PN) firing-rates, the precise functional role of synaptic-depression at the ORN synapses is not yet fully understood. In this paper we propose two hypotheses linking the information-coding properties of the fly AL with the network mechanisms responsible for ORN-->AL synaptic-depression. Our first hypothesis is related to variance coding of ORN firing-rate information--once stimulation to the ORNs is sufficiently high to saturate glomerular responses, further stimulation of the ORNs increases the regularity of PN spiking activity while maintaining PN firing-rates. The second hypothesis proposes a tradeoff between spike-time reliability and coding-capacity governed by the relative contribution of vesicle-depletion and presynaptic-inhibition to ORN-->AL synaptic-depression. Synaptic-depression caused primarily by vesicle-depletion will give rise to a very reliable system, whereas an equivalent amount of synaptic-depression caused primarily by presynaptic-inhibition will give rise to a less reliable system that is more sensitive to small shifts in odor stimulation. These two hypotheses are substantiated by several small analyzable toy models of the fly AL, as well as a more physiologically realistic large-scale computational model of the fly AL involving 5 glomerular channels

A schematic of the large-scale network model.

<p>[Left]: The network consists of 5 glomerular channels, each incorporating 60 olfactory receptor neurons (ORNs in green) which stimulate a ‘glomerulus’ consisting of 6 projection neurons (PNs in red), 6 excitatory local neurons (LNEs in magenta) and 6 inhibitory local neurons (LNIs in blue). The PNs, LNEs and LNIs are connected to one another randomly within each glomerulus, and the LNEs and LNIs also affect the neurons in other glomeruli. The LNIs affect the ORNAL synapses via presynaptic-inhibition. [Right]: The non-negligible connection strengths are listed on top, with the slow-inhibitory connection strengths listed separately from the fast-inhibition strengths. The relevant connection probabilities are listed on the bottom. The parameter refers to , which characterizes the overall strength of presynaptic-inhibition. See Methods for full details.</p

PNs exhibit broader odor responses than their associated ORNs.

<p>[A] Shown are trial-averaged firing-rate curves for various model PNs (magenta, solid) and associated model ORNs (green, dashed) in response to various model odors. Note that, qualitatively similar to experiment <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002622#pcbi.1002622-Olsen2" target="_blank">[12]</a>, the activity of the model PNs does not necessarily reflect the activity of the associated model ORNs. [B] Shown are the PN-ORN (green) and PN-PN (red) Spearman rank-correlation histograms for the model PNs and associated model ORNs (averaged over all PN and ORN pairs associated with each given glomerulus, and then further averaged over glomeruli — see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002622#pcbi.1002622-Bhandawat2" target="_blank">[20]</a> for the statistical methods used). Note that, qualitatively similar to experiment, the mean of the PN-ORN histogram is closer to than the mean of the PN-PN histogram, indicating that, while PNs associated with a given glomerulus tend to respond to the same odors, they do not necessarily respond to the same set of odors which stimulate their associated ORNs.</p

A simple analyzable cartoon of the tradeoff between reliability and sensitivity.

<p>In this example , and . In panels A and B the vesicle-depletion parameter . In panels C,D,E and F, the vesicle-depletion parameter , such that the mean firing rate is held constant. [A] Graphs of (solid), (dashed), (gray), and (gray dashed), as functions of , for the case . [B] Graphs of var (solid) and var (dashed) as functions of , for the case . [C] Graph of as a function of , subject to the constraint that remain constant. The constant value of chosen (essentially arbitrarily) in this case is the value of shown in panel A for . Other choices of yield similar results. Note that this graph is monotonically decreasing, implying the existence of a 1-parameter family of networks possessing the same — ranging from type-A networks with low and high , to type-B networks with high and low . [D] Graphs of (solid), (dashed), (gray), and (gray dashed), for the case . [E] Graphs of var (solid) and var (dashed) as functions of , for the case . [F] Graph of the optimal choice of (implying a vesicle-depletion parameter of ) for which discriminability is maximized, as a function of the sample number . The notion of discriminability is described in the section entitled “A simple cartoon of optimizing discriminability over short observation-times”. In this case the observation error is fixed at . Note that for low , discriminability is maximized for a type-B network. However, as increases, discriminability is maximized by type-A networks. The graph shown plots for , as for this particular simple example the derivative of reaches a vertical asymptote at .</p

An example of subnetworks which come into play when considering the sensitivity or reliability of the LNI.

<p>On the left in panel-A we show a particular network, with various ORN-LNI pairs (shown as ovals and circles respectively) connected via presynaptic-inhibitory connections. We will adopt the convention that the ORN-LNI pair is fixed (highlighted in dark gray), whereas the indices are not fixed, but are considered distinct from and from each other. Several dynamic features associated with the LNI can be determined by considering an expansion of the dynamics of this full network in terms of subnetworks. Shown on the right in panels-B,C,D are -order, -order and -order subnetworks of the full network which are relevant for determining the sensitivity and reliability of the LNI. The -order subnetwork consists of the ORN-LNI pair alone. The two -order subnetworks shown are those incorporating a single presynaptic-inhibitory connection — namely (top) and (bottom). The full network has embedded within it three -order subnetworks of the form , and one -order subnetwork of the form . The five -order subnetworks shown are those incorporating two presynaptic-inhibitory connections. Listed in reading order, these subnetworks are denoted by , , , , and . The full network has embedded within it , , , and of these subnetworks, respectively.</p

PNs are more reliable than their individual ORN inputs.

<p>Shown are averaged response curves for a typical model PN (magenta, solid) and model ORN (green, dashed) associated with the same glomerulus in our model. The grey overlay indicates the odor presentation period. Spikes were counted in bins. The mean spike-count per bin (averaged over trials) is shown on the left. The standard-deviation in spike-count per bin is shown in the center, and the coefficient of variation (standard deviationmean) is shown on the right. Note that, qualitatively similar to experiment <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002622#pcbi.1002622-Bhandawat2" target="_blank">[20]</a>, the model PN activates more quickly, has higher firing-rates, and is more reliable than the ORN.</p

Presynaptic-inhibition is partly responsible for ORN PN synaptic-depression.

<p>Shown is a scatterplot displaying the correlation between total ORN activity across all glomeruli in response to various odors, and the suppression of spontaneous EPSPs associated with a particular PN associated with a glomerulus which has been ‘shielded’ (i.e., the odor stimulus chosen does not affect the input drive to that glomerulus). In analogy with <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002622#pcbi.1002622-Olsen2" target="_blank">[12]</a>. The PN suppression is measured as the difference in integrated PN membrane potential between (i) the scenario in which the PN receives spontaneous spikes from its associated ORNs in the absence of any odor, and (ii) the scenario in which the glomerulus associated with that PN is shielded and an odor is presented, in which case the activity generated within the other glomeruli reduce the effect of the spontaneous spikes impingent on the PN, and the spontaneous EPSPs are absent or greatly diminished. Note that, due to presynaptic-inhibition within the model, the correlation between PN EPSP magnitude and total ORN activity is qualitatively similar to experiment <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002622#pcbi.1002622-Olsen2" target="_blank">[12]</a>.</p

A manifestation of variance coding within the large-scale model.

<p>The large scale model (described in Methods) exhibits a phenomenon similar to the variance coding shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002622#pcbi-1002622-g002" target="_blank">Fig. 2</a>. We constructed a panel of 16 odors, all of which only directly stimulated the same glomeruli (although to differing degrees). Moreover, we chose every odor within this panel such that the ORN firing-rates of the directly stimulated glomeruli were sufficient to saturate the firing-rates of the associated PNs (i.e., the directly stimulated ORN firing-rates were 12 Hz, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002622#pcbi-1002622-g010" target="_blank">Fig. 10</a>). Given this panel of odors, we presented each odor multiple times, and used the collection of -component PN firing-rate vectors (measured over the period immediately following odor onset) to perform a variety of odor discrimination tasks (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002622#s2" target="_blank">Results</a> for details). [A] The histogram of discriminability rates associated with -way discrimination tasks when only firing-rate data is used. Note that is chance level for these tasks (chance level is also shown in panels B,C,D). [B] The histogram of discriminability rates associated with the -way discrimination tasks when only firing-rate data is used (note that is chance level for these tasks). [C] The histogram of discriminability rates associated with -way discrimination tasks when firing-rate data and -point correlations (correlation time ) are used. [D] The histogram of discriminability rates associated with -way discrimination tasks when firing-rate data and -point correlations (correlation time ) are used. Note that the typical discriminability rate is higher when correlations are used. [E] Here we plot the difference in mean discriminability for the -way discrimination task between the cases (i) when firing-rate data and -point correlations are used, and (ii) only firing-rate data is used. We plot this difference as a function of the parameters and used in our large-scale model. The vesicle-depletion parameter ranges from to across the vertical axis, and the presynaptic-inhibition parameter ranges from to across the horizontal axis. The data shown in panels A–D is taken from the simulation indicated by the dashed square. Note that, as the total amount of synaptic-depression decreases, the discriminability computed using only firing-rates is closer to the discriminability computed using both firing-rates and -point correlations. [F] Similar to panel-E, except for the -way discrimination task, rather than the -way discrimination task.</p