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The final part develops a model that incorporates both the agglomeration and transport function of the Tat system, thereby providing a comprehensive description of this self-organizing process. Skip to main content Skip to table of contents. Advertisement Hide. Front Matter Pages i-xii. Pages Assembly and Fragmentation of Tat Pores.

About this book Introduction With the aim of providing a deeper insight into possible mechanisms of biological self-organization, this thesis presents new approaches to describe the process of self-assembly and the impact of spatial organization on the function of membrane proteins, from a statistical physics point of view. Macromolecular dynamics Mechanosensitive channels Membrane functional organization Outstanding PhD thesis Protein assembly Protein-protein interactions Self-organization and self-assembly.

What is the nature of such protein waves? In conventional reaction-diffusion systems e. Here real wave is driven by diffusion—a real propagation of chemicals in physical space 1. In contrast, pseudowave is not a real material propagation in space; instead, it reflects the spatial gradient in the timing of local excitation that gives the propagating impression.

The question is: in the context of chemical traveling waves, is our cortical protein wave propagation more akin to real or pseudowave? To gain insight in cortical protein traveling waves, we carried out more vigorous mathematical analysis of our system. By ignoring protein lateral diffusion along the cortex, we simplified the model to make it possible to obtain analytical solution of traveling wave. A nontrivial steady-state traveling wave solution emerged with a nonzero membrane curvature at the wavefront.

Interestingly, without protein lateral diffusion, this analytical solution yielded similar wave speed as those from the full model Fig. This finding suggests that diffusion need not be important in wave propagation, indicating that our protein traveling wave may not reflect real material movement in the direction of wave propagation. To further pinpoint the nature of our protein traveling wave, we borrowed the criteria that distinguishes pseudowave from real wave 49 : a wave is a pseudowave if the concentration changes—resulting from temporal chemical reaction at each point in space—are much larger in magnitude than those resulting from diffusion.

In this regard, diffusion in our nominal case contributed little but negatively to the wave propagation over time Fig. Taken together, we suggest that our cortical protein traveling wave is more in line with pseudowave. Importantly, this is the only type of cortical protein traveling wave in our system—a unique feature distinct from conventional reaction-diffusion systems—that typically host both real and pseudowaves. Curvature sensing-mediated cortical protein waves always reflect protein recruitment from cytoplasm onto membrane.

The analytic result was obtained by computing the analytical solution Eq. We note that there is one parameter in the analytic formula that cannot be analytically derived from the reaction rates in the full model. And we chose the location ahead of the wavefront, where the absolute value of membrane curvature is the highest.

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Variation in this location does not change the qualitative conclusions from these plots. We emphasize that the comparison here is only within the regime of traveling waves, not stationary wave phenomena that could emerge with characteristic spatial periodicities from conventional reaction-diffusion systems. The experimental data were collected from 37 waves in 8 cells.

The F-BAR intensity for each wavefront was normalized to the maximum value within the individual cell. The gray shade represents the tracking measurement uncertainty Supplementary Figs. We then asked: what underlies this unique feature? While all traveling waves must have the promotion zone ahead of the wavefront to confer propagation Fig. In conventional reaction-diffusion systems, as lateral diffusion advances the wavefront, autocatalytic reactions of the local chemicals always promote excitation in the direction of wave propagation, because the chemical concentrations are always positive Fig.

In contrast, the stimulatory elements in our system include not only conventional chemical autocatalytic reactions but also curvature sensing effects. The membrane curvature changes sign from negative to positive as one moves away from the wavefront in the direction of wave propagation Fig. Because F-BAR only accumulates in the negative-curvature region, it is not recruited to the positive-curvature region due to the geometric mismatch. As such, the curvature sensing constrains the effect of protein diffusional drift on wave propagation Fig.

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We therefore suggest that curvature sensing-mediated traveling waves of cortical proteins mainly reflect the local protein recruitment from cytoplasm, rather than the protein lateral diffusion Fig. What really propagates here is the membrane undulation that induces local protein recruitment from cytoplasm, potentiating wave propagation like a ripple moving in a pond. For the corresponding cortical protein traveling wave, it is the spatial gradient in the timing of local excitations that gives the propagating impression.

Hence, the smaller this spatial gradient, the smaller the relative timing in excitations and, hence, the faster the protein wave appears to propagate. To test this prediction, we used TIRFM experiments to track wavefront propagation with high spatial-temporal accuracy. For a given wave, wave speed is heterogeneous. Importantly, our experiments show that the spatial gradient of cortical protein density at the wavefront inversely correlates with the wave speed circles, Fig.

Because the spatial gradient of cortical proteins reflects the relative timing of the local excitation Fig. More critically, a zoom-in kymograph shows that the individual FBP17 punctum undergoes cycles of assembly and disassembly during wave propagation, but do not notably move in space Fig. We conclude that our cortical protein traveling waves mainly reflect the protein recruitment from cytoplasm, rather than, the lateral diffusional drift. In this work, we provide some mechanistic insights into cortical traveling waves. Our work highlights the importance of membrane shape change in establishing a mechanical cue for cortical protein recruitment, which reciprocally governs cortical dynamics that culminate in traveling waves.

Critically, we show that in a curvature sensing-driven rhythmic propagation, the cortical protein traveling wave results mainly from the local protein recruitment from the cytoplasm, rather from lateral diffusion. Our cortical protein traveling waves have several unusual features. This is because the protein traveling waves emerge from the membrane shape-mediated feedback, and are constrained by the accompanying membrane undulation that itself is a real wave.

Second, while the speed of our mechanochemical wave is ultrafast, it is still modulated by the protein lateral diffusion along cortex. This is different from another interesting example of a mechanochemical wave 53 , whose predicted wave speed is independent of protein lateral diffusion, as this chemical wave is entirely driven by and, hence, effectively reads out, membrane mechanics In contrast, in our model wave speed is controlled by the feedback between membrane curvature and cortical protein dynamics, and is not completely independent of protein lateral diffusion Fig.

Third, in our model the sub-diffusive dependence of wave speed on protein lateral diffusion constant is a consequence of curvature sensing, not just because of the mechanochemical feedback. To demonstrate this, we altered the model in several ways, maintaining mechanochemical feedback but without curvature sensing, and found that the corresponding wave speed was not always sub-diffusive Supplementary Fig. In contrast, the wave speed in the other model schemes—that preserve curvature sensing in the feedback—is always sub-diffusive Fig.

Interference between traveling waves could make it possible for cells to coordinate and integrate cortical signals.


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While our model focuses on a specific system, it provides a general framework for curvature sensing-mediated traveling wave dynamics. The unusual features remain robust to variations in the detailed model scheme. They only require autocatalytic cortical recruitment of a curvature-sensitive protein coupled with a negative feedback possibly involving actin dynamics Supplementary Fig. Given that there are many other curvature-sensitive proteins involved in diverse cellular processes 5 , 18 , 54 , 55 , 56 , the biological implications of our model can be broad.

We note that curvature sensitive I-BAR domain proteins e. Our model recapitulates the basic features of our traveling wave but is inevitably incomplete. First, we focused on the cortex-centric mechanism of rhythmic propagation. An open question is: can the cortex-bound protein wave be part of a real wave in the cytoplasm? Even if the wave exists in the cytoplasm, we reason that the membrane shape change and the curvature sensitivity of F-BAR are still essential for this rhythmic propagation Figs.

Future work will investigate the possibility of traveling waves with a cytosolic origin. Second, the model treated the ventral membrane at the cell edge as a clamped boundary. While external stimulation i. This is because the membrane at the epicenter does not entirely relax back to the baseline after the protein wave has passed. It is this residual membrane shape deformation that serves as a cue to recruit F-BAR, which in turn initiates the next round of oscillation Supplementary Fig.

On the other hand, because the clamped boundary dampens the membrane shape changes, its effect propagates from the edge to the epicenter, where it flattens the residual membrane shape deformation over time. Without external activation signals, this flattening of the membrane eventually prevents a new round of F-BAR cortical recruitment and, hence, dampens the oscillation at long times.

This is consistent with our observations that the waves are only on the cell ventral side and not strictly sustained oscillations as indicated by Fig. Nevertheless, our key conclusion remains robust regardless whether the dynamics is a sustained or dampened oscillation. In reality, the cell edge may evolve over time; and membrane-substrate adhesion is probably not spatially uniform as assumed.

Interestingly, a localized adhesion that clamped the membrane diverted the wave propagation Supplementary Fig. In the future, we would like to systematically investigate how dynamics of cell edge movement and cell-substrate adhesions impact our traveling wave. Third, our model proposed that the membrane stiffening by actin accumulation prevented membrane deformation and hence turned off F-BAR recruitment. This is consistent with a negative role of actin in cortical wave observed previously 3 , and supported by our observations that the actin accumulation negatively correlates with the membrane height change Supplementary Fig.

Also, latrunculin A treatment has effects on traveling waves similar to hyper-osmolarity Fig. However, this proposal does not exclude additional effects of actin. For instance, actin polymerization may push the membrane toward the substrate, increasing the membrane-substrate adhesion. We showed that our key conclusion was robust to this model variation Supplementary Fig. Alternatively, actin retrograde flow could remove F-BAR from the membrane. Dissecting more detailed molecular events of actin machinery in our traveling waves will be part of our future work.

In sum, our work identifies membrane shape change as an important factor in determining the dynamics of traveling waves propagating along the cell cortex. This finding warrants closer scrutiny of the role of curvature sensing in other rhythmic cortical phenomena 2 , 3 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , possibly including actin-based cell polarity establishment 58 , 59 , 60 , 61 and cytokinesis For transient transfections, electroporation with Neon transfection system Life Technologies was used.

After transfection, cells were plated at subconfluent densities in mm glass bottom dishes MatTek, Ashland, MA or on round coverslips in a well plate overnight. In all experiments, cells were sensitized with mouse monoclonal anti-2,4-dinitrophenyl IgE Sigma-Aldrich at 0.

Optimised Projections for the Ab Initio Simulation of Large and Strongly Correlated Systems

DNA sequences corresponding to a. Constructs for the following proteins were kind gifts: Lifeact-mRuby from Dr. All plasmids were sequenced to vindicate their integrity. MetaMorph 7. Reflected light rays from two interfaces between coverslip and solution and between solution and plasma membrane interfered with each other, creating a bright or dark patch when they were constructive or destructive, respectively.

Prior to imaging, the coverslip was transferred to a custom perfusion chamber Chamlide, LCI placed on the heated stage of a Nikon Ti-E inverted microscope. A multi-valve perfusion control system MPS-8, LCI was used to switch rapidly between solutions flowing into the chamber and over the cells. The perfusion system was connected to a computer and controlled by MetaMorph software Molecular Device in synchronization with image acquisition.

Co-immunoprecipitation Protocols and Methods | Springer Nature Experiments

An ImageJ-based software Fiji 63 was used to generate movies, kymographs, montages, and Z-stack projections. We determined the instantaneous wave speed and the cortical protein gradient at wavefront by the following procedures as illustrated in Supplementary Figs. To avoid complications arising from the interactions between multiple waves, we only chose well-defined single waves to determine the wave speed and the cortical protein gradient at the wavefront. We set the boundaries of individual waves by determining where the image intensity fell to below a threshold slightly above the background intensity level Supplementary Fig.

Julie Theriot (Stanford, HHMI) 3: Evolution of a Dynamic Cytoskeleton

Next, the wave centroid r c t i of the segmented region for each time frame t i was calculated Supplementary Fig. The line intensity profile of FBP17 along the wave propagation direction the arrow in Supplementary Fig. Here I rc was the intensity at centroid, I bk was the background intensity, I max was the maximum intensity of this measured cell, and d g was the distance from wave centroid to the wave edge where the intensity decreased to the background level Supplementary Fig.

The robustness of our wave-tracking results depended on the chosen segmentation threshold value. To ensure that the tracking can give consistent results, we varied segmentation threshold values to determine the optimal range. Our calculation showed that at each time point of a given wave, the instantaneous wave speed from our tracking analysis varied with the segmentation threshold step 1 in Supplementary Fig. We chose the segmentation threshold to be in a range within which the speed variation was minimal step 2 in Supplementary Fig.

By overlapping the individual range from every wave at every time point of the same cell, the overall optimal threshold range for this cell was determined steps 3 and 4 in Supplementary Fig. This optimal range could vary in different cells in part due to the different background intensities.

Throughout the tracking, we thus chose a fixed segmentation threshold value within the optimal range that is independently determined for individual cells. Further, to calculate the FBP17 gradient, we obtained its intensity profile along the wave propagation direction by averaging its intensity value around centroid within the box step 1 in Supplementary Fig. Consequently, the measured FBP17 gradient was in subject to the box size step 2 in Supplementary Fig.

For each time point of an individual wave, the variation in the measured FBP17 gradient using different box sizes was then determined step 3 in Supplementary Fig. By calculating all the gradient variations from different waves at different time points, we obtained the correlation between the uncertainty and the average of our measured gradient steps 4 and 5 in Supplementary Fig.

Importantly, the smaller the average gradient was the smaller was the uncertainty. The uncertainty in the FBP17 gradient measurement was plotted as the gray shadow in Fig. Data supporting the findings of this study are available within the article and its Supplementary Information file and from the corresponding authors on reasonable request.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Winfree A. The Geometry of Biological Time , 2nd edn. Springer Allard, J. Traveling waves in actin dynamics and cell motility.

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Optical Properties of Nanostructured Metallic Systems

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