Relating trajectories of uniform and variable yield populations.

<p>All gray trajectories, ending on the solid red line (variable-yield stopping line) at points corresponding to , build up the cumulative probability for the final population to have less than cells of type 1. Due to the monotone property of trajectories, they all cross also the dashed line (uniform-yield stopping line) that passes through the point obeying the same ocnstraint and thus the cumulative probability is the same. The parameters of the two lines are simply related through (see Eq. (11)).</p

Final population size for a heterogeneous micro-population with metabolic tradeoff.

<p>(A) Distributions of the final populations size from simulations with division rate ratio and yield ratio , for different initial population sizes - cells (solid line), cells (dashed line), cells (dotted line), cells (dash-dot line). In (B) we can see the Standard deviation of the final population size as function of initial population size, in good agreement with the analytic approximation.</p

Average final vs. initial population size in micro-populations grown to saturation of resource.

<p>Dotted line: a population with a uniform yield. Symbols: Monte Carlo results for two-state populations with variability in yield and in growth rate. Dashed lines: analytic approximations relevant only for special parameter values. Crosses: Monte Carlo simulation for “metabolic tradeoff” (lower crosses), (upper crosses). circles: variable yield positively correlated with division rate (upper circles), (lower circles).</p

Trajectories in phenotypic space for heterogeneous micro-populations grown to saturation of resource.

<p>Each point in the plane represents the number of cells of each type in the population. Division of type 1 increases by one and corresponds to the trajectory advancing along the -direction, and similarly for type 2. All three panels show trajectories that start from an initial population of . (A) symmetric types (, ). Trajectories are equally likely to proceed along or , and the stopping condition is of slope (). (B) Types differ by their division rates (, ), causing the trajectories to be biased towards the faster growing type. (C) Types differ in their yield with respect to the finite resource (, ), causing the stopping line to be of slope different from (); growth will stop after a variable total number of divisions depending on trajectory, since each type consumes a different amount of resource at division.</p

Scaled distribution of the number of cells of metabolic type 1 in the final population.

<p>All distributions are for symmetric initial composition, equal yields and a large number of divisions. Different distributions in a plot are for different initial populations (Blue - , Green - , Red - ). These distributions are plotted as a function of the scaling variable (see text for details), and their shape does not depend on the number of divisions but does depend on the initial number of cells. (A) The two types have the same growth rate and are therefore equal in all their properties. Population composition varies only because individual trajectories are composed of different sequences of divisions of the two types. Because of the symmetry between types, all distributions are symmetric around . (B) The two types have different growth rates, and the distribution of final composition becomes skewed.</p

Cooperative stochastic binding and unbinding explain synaptic size dynamics and statistics

<div><p>Synapses are dynamic molecular assemblies whose sizes fluctuate significantly over time-scales of hours and days. In the current study, we examined the possibility that the spontaneous microscopic dynamics exhibited by synaptic molecules can explain the macroscopic size fluctuations of individual synapses and the statistical properties of synaptic populations. We present a mesoscopic model, which ties the two levels. Its basic premise is that synaptic size fluctuations reflect cooperative assimilation and removal of molecules at a patch of postsynaptic membrane. The introduction of cooperativity to both assimilation and removal in a stochastic biophysical model of these processes, gives rise to features qualitatively similar to those measured experimentally: nanoclusters of synaptic scaffolds, fluctuations in synaptic sizes, skewed, stable size distributions and their scaling in response to perturbations. Our model thus points to the potentially fundamental role of cooperativity in dictating synaptic remodeling dynamics and offers a conceptual understanding of these dynamics in terms of central microscopic features and processes.</p></div

Synaptic size dynamics in the bidirectional cooperativity model.

<p><b>(A)</b> Sizes of 5 simulated synapses over 50 simulation time steps. Note the fluctuations in simulated synapse size over the course of the simulation. <b>(B)</b> Scatter plot of changes in synapse size as a function of initial size after 50 time steps (3,500 synapses). <b>(C,D)</b> Changes over time of synaptic sizes for the same synapses after 5 (C) and 50 time-steps (D). Dashed red lines represent linear regression fits, with fit coefficients shown in the figure. <b>(E)</b> Slopes and offsets of linear regression lines in plots such as those of C and D for 300 consecutive time steps. Offsets were normalized to mean synaptic size during this 300 time-step window. <b>(F)</b> Coefficients of determination (R²) in plots such as those of C and D for 300 consecutive time steps. All data were obtained after mean synaptic size plateaued at ~225 bound molecules (after about 900 steps). See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005668#sec013" target="_blank">Methods</a> for simulation parameters.</p

Stable stationary distributions in the global cooperativity model.

<p>Solutions of the global bidirectional cooperativity model, obtained by the Fokker-Planck approximation to the master equation (for details, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005668#pcbi.1005668.s005" target="_blank">S1 Appendix</a>, Section 2). (A) Distribution of synaptic sizes scanning a range of cooperativity parameter C (right-hand axis). For C<<1, we find stationary distributions very close to Gaussian (orange distributions). In this limit we approach a situation similar to the Langmuir model. For C>>1, when cooperative processes dominate, skewed distributions are found as observed experimentally (blue distributions). (B) Same stationary distributions as in (A), but scaled by subtracting the mean and dividing by the standard deviation. For technical reasons, a weak non-cooperative unbinding rate β was added (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005668#pcbi.1005668.s005" target="_blank">S1 Appendix</a> for a detailed justification); here β = α, and other parameters as in Methods.</p

Synaptic size distributions for the Langmuir model.

<p><b>(A)</b> Illustration of independent binding and unbinding from the matrix. <b>(B)</b> Synaptic size distributions (semi-logarithmic scale) obtained from the Langmuir model for three values of <i>k</i>_<i>on</i>, with <i>k</i>_<i>off</i> set to 0.5. Parabolic shape of the curves corresponds to Gaussian-like distributions. Simulated data for 3,500 synapses. See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005668#sec013" target="_blank">Methods</a> for the rest of the simulation parameters used here.</p

Synaptic size distributions in the Contact Process model.

<p><b>(A)</b> Illustration of the Contact Process: Binding is cooperative with a rate that increases with the fraction of occupied neighboring sites (representing interactions with nearby bound molecules) whereas the unbinding rate is a constant, β, insensitive to the numbers of occupied neighboring sites. Binding is also affected by a small, constant and non-cooperative component (<i>α</i>) representing weak, non-specific binding to the matrix. <b>(B)</b> Synaptic size distributions for different λ_on/β ratios. All distributions were determined numerically through simulations (3500 synapses, 1500 time steps; see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005668#sec013" target="_blank">Methods</a> for further details). Note the semi-logarithmic scale. <b>(C)</b> Same distributions as in (B) after scaling by subtracting the mean and dividing by the standard deviation. See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005668#sec013" target="_blank">Methods</a> for the rest of the simulation parameters used here.</p