Modeling the Kinetics of Open Self-Assembly

In this work, we explore theoretically the kinetics of molecular self-assembly in the presence of constant monomer flux as an input, and a maximal size. The proposed model is supposed to reproduce the dynamics of viral self-assembly for enveloped virus. It turns out that the kinetics of open self-assembly is rather quantitatively different from the kinetics of similar closed assembly. In particular, our results show that the convergence toward the stationary state is reached through assembly waves. Interestingly, we show that the production of complete clusters is much more efficient in the presence of a constant input flux, rather than providing all monomers at the beginning of the self-assembly

Radii distribution of the truncated spheres-MLR estimated from the experimental PALM-data consisting in <i>n</i> = 33 VLPs.

<p>The mean value as well as intervals of ±1 and ±3 the standard deviation are shown (red plain line and red dashed lines respectively). Each VLP contribution is a normalized gaussian centered on the estimated radius and whose variance is given by the inverse of the observed Fisher matrix. The second peak <i>R</i> ≃ 90 nm is due to a particle that is likely an aggregate (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0172943#pone.0172943.s006" target="_blank">S5 Fig</a>) and is not taken into account in the mean and variance.</p

Maximum Likelihood Reconstruction efficiency on completion.

<p>Estimated completion values (zenithal angles) of 300 simulated superresolution data-sets determined by the MLR method are plotted against ground truth values (symbols). Particles for which the planar orientation angle <i>α</i> is ill-determined (quasi-isotropic distribution with Δ<i>α</i> > <i>π</i>/10 -see Eq S9 in supplementary information <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0172943#pone.0172943.s001" target="_blank">S1 File</a>) are shown as empty triangles. An ideal estimator would give <i>θ</i>_measured = <i>θ</i>_real (red dashed line). Calculations of the Fisher information matrix are used as estimates of the precision reached by the estimator for each VLP (grey error bars). For the figure to be readable, plotted error bar are only a third of the estimated std.</p

Distribution of estimated completion angles given by the MLR method on the experimental PALM-data (<i>n</i> = 33 VLPs—blue), mean value and interval of ±1 and ±3 the standard deviation are shown (red plain line and red dashed lines respectively).

<p>Each VLP contribution is a gaussian centered on the estimated radius and whose variance is given by the inverse of the observed Fisher matrix. As acceptable completion angle lay in [<i>π</i>/2, <i>π</i>], only this part of the gaussian is considered and normalized.</p

Fitted structure and simulated data.

<p>(A) The assumed density of Gag (transparent blue) is shaped as a truncated spherical shell of radius R, completion angle <i>θ</i>, and is tilted by an angle <i>ϕ</i> with the projection axis (the optical axis). Its projection is distributed around the projected sphere center (<i>x</i>_<i>c</i>, <i>y</i>_<i>c</i>) and makes an angle <i>α</i> with the plan <i>x</i>-axis. (B) To simulate palm images, we uniformly sample points in this density, project them on the plane (red dots), and move each of them by a random normal displacement of std. <i>σ</i>_<i>i</i> (black dots) accounting for the superresolution imprecision.</p

Biochemical characterization of VLPs and viral cores, and AFM imaging.

<p><i>(a)</i> Immunoblot of HIV-1 for mature and immature VLPs and cores. <i>(b)</i> Reverse transcription test on mature VLPs and cores in the presence or the absence of ψRNA in the VLPs or cores. <i>(c)</i> Typical images of mature VLPs. <i>(d)</i> Typical images of viral cores.</p

Automated image analysis.

<p>(a) Example of the successive image analysis steps applied on a single image. (b) Cartoon of the principle of image analysis. (c) Various representation (color map, contour plots, 3D plots) of particles that were selected. <i>Top</i> VLP, <i>down</i> core. Interestingly, the contour map of the VLP shows a pronounced asymmetry close to its maximal height, reflecting the presence of an asymmetric object inside the VLP.</p

Model of entropic selection of viral genome at fixed particle size.

<p><i>(a)</i> Cartoon of the self-assembly. The model is specialized to bimodal products of self-assembly, in which the size of particle are equal and the RNA content are different. (b) Typical large RNA titration computed thanks to the model detailed in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0083874#pone.0083874.s001" target="_blank">File S1</a>. The value of parameters chosen for the calculation are found in the <i>material and methods</i>. Blue circles correspond to particles with large RNA, and green crosses correspond to particles lacking large RNA.(c) Phase diagram <i>small RNA/large RNA</i>. The boundary for which the concentration of both particles are equal is shown by blue filled circles. The line joining the circles is drawn to guide the eye.</p

Influence of the presence or absence of viral ψRNA on particle morphogenesis.

<p>The number of particles is indicated by the value <i>N</i>. <i>(a)</i> and <i>(c)</i> 2D histogram of short and long diameters for respectively VLPs in the absence and in the presence of ψRNA. The size distribution is shifted toward smaller value, and its dispersion is reduced. <i>(b)</i> and <i>(d)</i> 2D histogram of short and long diameters for respectively cores in the absence and in the presence of ψRNA. The same shift in the distribution is observed, although with w weaker amplitude. <i>(e)</i> Box plot of equivalent diameters summarizing the previous results. The equivalent diameter is obtained by converting the 2D projected area of the particle into the diameter of a disk that would give the same area.</p

Combined model of viral genome and particle size entropic selection.

<p>The model is specialized to bimodal products of self-assembly, with different particle size and different RNA content. <i>(a)</i> Typical large RNA titration computed thanks to the model detailed in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0083874#pone.0083874.s001" target="_blank">File S1</a>. The value of parameters chosen for the calculation are found in the <i>material and methods</i>. In this case, the number of proteins in the “large particles” is twice the number of proteins in the small one. Red circles correspond to small particles with large RNA, and pink crosses correspond to large particles lacking large RNA. <i>(b)</i> Phase diagram <i>small RNA/large RNA</i>. The boundary for which the concentration of both particles are equal is shown by red filled circles. The line joining the circles is drawn to guide the eye. For comparison, the boundary at fixed particle sizes found in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0083874#pone-0083874-g005" target="_blank">figure 5b</a> is depicted by a blue dotted line.</p