Actin gel dynamics:Matthias Bussonnier

Cortical tension measured on liposome doublets

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Cortical tension measured on liposome doublets

Introduction

We have seen that in cells, the actin cytoskeleton is a key component to form structures such as the actin cortex, used to transmit forces and bring to cells their mechanical rigidity. In order to drive shape changes, cells regulate the mechanical properties of the sub-micrometer thick actin cortex, beneath the membrane [Clark et al. 13]. The actin cortex dynamics drives cell shape changes [Salbreux et al. 12] and the presence of the molecular motor myosin II plays a fundamental role in the tension of the acto-myosin cortex [Tinevez et al. 09]. The cortical tension can be measured on cells and varies between 50 and 4000 pN/µm in accordance with both actin and myosin activity .

These changes of the cortical tension are also affected by cell-cell adhesions [Maitre et al. 12] whose major role played in cell sorting have already been proved.

Recently, such acto-myosin cortices have been reconstructed on cell-sized liposomes [Carvalho et al. 13a], which showed that the attachment of the actin cortex to the membrane plays a crucial role in the acto-myosin network behavior and contractility.

In the present study, I collaborated with Kévin Carvalho and Joël Lemière, to further extend the previously developed system [Carvalho et al. 13a] aiming at monitoring the cortical tension changes in a biomimetic actin cortex formed on liposomes. I mainly contributed to the analysis of the 3D data, acquired by using Spinning Disk Microscopy and , for the analysis developed for this purpose a novel method to get a precise and unbiased measure of the geometrical parameter.

It should be noticed that in a recent works, cell doublets were used to determine the role of cortical tension in cells [Maitre et al. 12]. In the present case, we formed similar doublets from liposomes, around which we polymerised an actin cortex in vitro (Fig 461). The shape changes of these liposome doublets allowed the time-dependent monitoring of cortical tension in a non-invasive way. In this project, we hence developed a method for the precise acquisition of doublet deformation, in order to accurately determine the tension increase induced by the injection of myosin motor on the preformed actin cortex.

Experimental description

Formation of liposomes doublets

Liposomes were obtained by electro-formation (see Material and methods) from a mix of EPC and PEG-biotin lipids. The presence of streptavidin in the working buffer allowed liposomes to naturally stick together to form doublets after 15 minutes (Fig 461).

../_images/Fig_01-A.png

Figure 461: Cell-sized liposome doublets. Doublets are indicated by white arrows in the field of view of a phase contrast microscope.

Formation of actin cortex on doublets

The formation of the actin network on doublets was done in a similar way as recently described [Carvalho et al. 13a]. Briefly, actin filaments including biotinylated monomers were stabilised by phalloidin and linked to PEG-Biotin lipids (see materials and methods) via streptavidin, present in the solution (Fig 463). Besides linking the actin to the membrane, it also cross-linked the filaments. Such a network has already been recently characterised [Carvalho et al. 13a]. Note that as the actin filaments were only added after the formation of the doublets, the interface between the two liposomes composing the doublets remained free of F-actin (Fig 464, 462). As the added actin was fluorescent, the absence of actin at the liposome interface could be checked by epifluorescence, as it appeared dark compared to the rest of the doublet(Fig 464).

Formation doublet schema

Figure 462: Formation of doublets: 1) In the presence of streptavidin, single liposome (A) aggregates into doublets. (B) The addition of biotinylated actin filaments stabilized with phalloidin (2) forms liposome doublets covered with a micrometer-sized actin network (C). The interface between the two liposomes is a double lipid bilayer free of actin filaments.

../_images/Fig_01-B.png

Figure 463: Schematic of the stabilized actin cortex at the membrane (proteins not to scale).

Visualisation of the interface

../_images/Fig_01-C.png

Figure 464: i) Flow-chamber designed for buffer exchange. Doublets are visualised in the middle horizontal channel of the H-shaped chamber to avoid movements during the buffer exchange. Spinning disk images of the doublet before i) or after iii) myosin II injection. One liposome contains the fluorophore SRB (red) to visualise the doublet interface. The actin cortex is labeled in green. Scale bar 5µm.

In order to visualise the interface between the liposomes, and to avoid the use of fluorescent lipids that might affect the membrane mechanics, [Sandre et al. 99] the inside buffer of approximately half of the liposomes was labeled with 0.9 µM of sulphorhodamin B (SRB eventually leading to half of the doublets containing a single fluorescent liposome (Fig 464 i and iii).

Geometrical parameters

To study the doublet geometry, we modelled each liposome and the interface between them as two spherical caps with their respective center and radius, as sketched in figure 465.

../_images/notations-doublets.png

Figure 465: The parameters notation for the doublet model: \(R_1\), \(R_2\), \(R_i\) are respectively the radius of the liposome 1, the liposome 2 and the interface. \(d\) is the distance between the liposomes centers. \(\theta_1\) and \(\theta_2\) are the angles between the tangents of the liposome surface and the tangent to the interface at the contact line. The total contact angle \(\theta\) is the sum of \(\theta_1\) and \(\theta_2\).

The center position in 3D (X,Y,Z) and the radius (R) of the three spherical caps completely determine the doublet geometry, though it is interesting to consider the other parameters of the doublets, which are :

  • the total volume of the liposome doublets V
  • the contact angle between the two liposomes
  • Every “half”-contact angles which are the angles between the interface and each liposome \(\theta_1,\theta_2\)
  • The distance between the liposome centers.

Experimental Observations

Effect of myosin-II injection

We imaged the liposomes doublets in an open chamber either in phase contrast and epifluorescence, or spinning disk microscopy in the red (sulphorhodamin) and the green (actin) channel.

The muscle Myosin II that formed bipolars filaments was carefully injected into the chamber, and led within a few minutes to a shape change (Fig 466) of the doublets, due to the actin cortex contraction.

../_images/doublet-contract.png

Figure 466: Doublets contraction showing a green channel(actin): (A) doublet before myosin II injection. (B) doublet during contraction due to myosin II. Time=0 corresponds to myosin II injection. Scalebar is 5 µm

The distance between the liposome centers decreased as the total angle \(\theta = \theta_1+\theta_2\) increased. The contact angle and other doublets parameters were obtained by fitting spherical caps onto the 2D epifluorescence images or on the 3D confocal stack as described later. In the absence of myosin, the contact angle \(\theta\) was measured to be \(\theta = 64 \pm 16 ^{\circ}\) (n=18), whereas in the presence of myosin II (200 nM) we found a value of \(\theta = 86 \pm 21 ^{\circ}\) (n=5). Measurements of the contact angle after myosin II injection were done before the cortex ruptures as characterised in [Carvalho et al. 13a].

Relation between the angles and tension

Each liposome has its respective tension \(\tau_1\), and \(\tau_2\). In the absence of the biomimetic acto-myosin cortex, these tensions only correspond to the tension of the liposome membrane. The interface between the two liposomes is formed by two lipid bilayers, and the inter-facial tension is composed of two contributions: the tension of the lipid bilayer, noted \(\tau_i\), and the adhesion energy per surface unit \(W\) due to the biotin-streptavidin-biotin link between the two lipid bilayers. The total tension at the interface can thus be written \(\tau_t = \tau_i -W\) [Maitre et al. 12].

As the movement of the contact line during the contraction is slow (order of µm/min) compared to the pressure equilibration across the doublet, we can consider the contact line between the liposomes and the interface to be at equilibrium. Hence, we can apply Young’s equation:

(1)\[\begin{split}\sum_{k \in interfaces} \tau_k. \vec t_k = \vec 0 \\ \tau_i \vec t_i + \tau_1 \vec t_1 + \tau_2 \vec t_2 + = \vec 0\end{split}\]

In which \(t_k\) are the vectors tangent to the interface at the contact point, as described in figure 467

../_images/yd.png

Figure 467: Equilibrium of the contact line. Each interface pulls on the line with a force proportional to its tension. As the contact line is at equilibrium, the sum of the forces compensate, thus ensuring a relation between the tensions and the contact angles.

This allows to relate the tension of all the lipid layers and the angle between them at each instance of the contraction. We can in particular project the result of this equation onto the direction of the contact surface tangent (dotted line on figure 467):

(2)\[\tau_i - W = \tau_1.cos(\theta_1) + \tau_2.cos(\theta_2)\]

And on the direction perpendicular to it :

(3)\[ \tau_1.sin(\theta_1) = \tau_2.sin(\theta_2)\]

These equations link the tension to the contact angle before, during and after the contraction and hence remain correct during the experiment. In the following, we will mark the values before the contraction phase by the suffix 0. Thus, for example \(\tau_{i,0}\) refers to the interface tension before the addition of myosin, and \(\tau_i\) refers to the interface tension at any instant of the contraction.

Contact angle dispersion

The value of the contact angle \(\theta\) varies across the different doublets both before and after the addition of myosin II. This reflects the initial variations of tension in \(\tau_{i,0}\), \(\tau_{1,0}\), and \(\tau_{2,0}\) from doublet to doublet. Such variations could be due to a difference in the liposome tension acquired during the different preparations, but also to a variation of adhesion energy between doublets, or alternatively to an effect of tension build-up during the actin shell formation. As the dispersion in the contact angle is in the same order as the increase in the angle upon addition of myosin, a statistical analysis of the contact angle before and during contraction is problematic. Thus, to avoid this effect of dispersion, we followed the evolution of \(\theta\) each individual doublet over time.

Tension of actin-shell

In order to investigate the tension increase due to the acto-myosin network on liposomes, we first characterised the increase, only due to the addition of the actin-shell in the absence of myosin. By destroying the F-actin via photo-bleaching (Fig 468) we compared the shape of the same doublets in presence and absence of the actin-shell. It should be noted that it is established that the actin filaments are destroyed by bleaching, as this process frees oxygen radicals that denature the actin monomers. Hence, the bleaching process actually destroys the actin cortex ([vanderGucht et al. 05]). This investigation showed that the total contact angle changes by \(3.4 \pm 2.0 ^{\circ}\) (n=7) after disruption (Fig 469) of the actin network. Thus, we concluded that the tension change due of the actin-shell is negligible compared to the tension change we can observe with myosin.

../_images/Fig_02-A.png

Figure 468: Image of an individual doublet coated with fluorescent F-actin before i) ii) and after iii) iv) actin cortex disruption. The actin cortex is visualised by epifluorescence ii) iv) and the doublet by phase contrast i) iii). Scale bar 5µm.

../_images/Fig_02-B.png

Figure 469: Measurement of the contact angle between the two liposomes forming the doublet before (black) and after (white) disruption of the stabilised actin cortex as a function of their volume.

3D observation

The three dimensional imaging of the doublets is necessary to get the correct contact angle. This requirement comes from the fact that in simple 2D epifluorescence images, the focal plane would have to correspond to the equatorial plane of the doublets for correct analysis. If this is not the case, the fit will produce a systematic underestimation of the contact angle. This is especially the case when doublets are of different radii, as typically found in our experiments, where the liposomes composing the doublets have an ratio of \(R_1 / R_2\) between 1.15 and 1.82.

../_images/light_table.png

Figure 470: Confocal stack of a liposome doublet actin channel, 3D reconstruction in figure 471. Note that there is no actin at the interface between the liposomes (Frames #11-#14). The distance between each image is \(\Delta z=0.85\) µm.

../_images/Fig_03-A.png

Figure 471: 3D reconstruction of a doublet surrounded by actin. The absence of actin on the interface can be more easily observed on figure 470.

Time resolved 3D Spinning disk stacks (Fig 470 with 3D reconstruction Fig 471) are recorded with a time resolution of less than 5 seconds per stack, for an accurate determination of the different doublet parameters over time. The analysis reveals the contact angle \(\theta\) (Fig 472) , the doublet volume \(V\) (Fig 474) and the distance between liposome centers \(d\) (Fig 473). All these parameters are obtained by fitting spherical 3D caps on the 3D stack as explained later.

../_images/Fig_03-B.png

Figure 472: Evolution of the contact angle compared to its initial value as a function of time. Each doublet is represented by a different color. The color code corresponds to the doublet shown in figure 473, 474 and 475. A special case is highlighted in the blue dashed line, where the actin cortex on the doublet ruptured, and the cortex is peeled off. The analysis of this case showed that the contact angle after rupture recovers its initial value.

../_images/Fig_03-C.png

Figure 473: Evolution of the distance between liposome centers as a function of time. Same color code for same doublets as in figure 472, 474 and 475. Again, the doublet with the ruptured cortex recovers its initial parameter values.

../_images/Fig_03-D.png

Figure 474: Evolution of the volume ratio over time. Same color code for same doublets as in figure 472, 473 and 475.

During the contraction triggered by myosin, we observed that the contact angle \(\theta\) increased while the distance between liposome centers \(d\) decreased. During this process, the volume remained constant within the error rate? of 10%. These results are consistent with the contact angle measures in freely adhering cell doublet experiments done previously [Maitre et al. 12].

Discussion

Cortical tension is homogeneous for single doublet

Combining the equation (3) with the finding that \(\theta_1 = \theta_2 = \theta /2\), allows to infer the equality of tension on both sides of the doublet during all the experiments. We can hence write \(\tau_1 = \tau_2 = \tau\). This result is consistent with the fact that actin is continuously distributed all around the liposome doublet. Hence, myosin II minifilaments pull on a continuous shell. In these conditions, equation (2) simplifies to :

(4)\[\tau_i - W = 2.\tau(t).cos(\theta(t)/2)\]

Where \(\tau(t)\) and \(\theta(t)\) are the tension and the angle at time \(t\) after myosin injection. That \(\tau_i-W\) may depend on a variability of the initial adhesion between liposomes. Since myosin does not operate at the interface between liposomes as it is actin-free, we can reasonably consider that both the tension and the adhesion energy are constant for a given doublet over time \(\tau_i-W = \tau_{i,0}-W_0\). Therefore, we obtain an expression of the tension \(\tau(t)\) during the acto myosin contraction that reads :

(5)\[\begin{split}\tau(t) &= \frac{ \tau_i - W }{2.cos(\theta/2)}\\ &= \frac{ cst }{2.cos(\theta/2)}\end{split}\]

Consequently, we can evaluate the tension relative to its initial value over time :

(6)\[\frac{ \tau(t) }{\tau_0} = \frac{cos(\theta_0/2)}{cos(\theta(t)/2)}\]

Relative increase in cortical tension

The interaction of myosin II filaments with a biomimetic actin cortex induces tension build-up. The cortical tension, normalised to its initial value, increases and reaches a plateau where \(\tau(t) = \tau_{peeling}\) (Fig 475), with the same trend as \(\theta\). Note that if the acto-myosin shell breaks and peels, the doublet recovers its initial shape (see dashed blue line for \(d\) and \(\theta\) on Fig 472, 473, 474 ). The average relative tension is found to be \(\tau_{peeling}/\tau_0 = 1.56 \pm 0.56\) (n=5) in 3D and \(\tau_{peeling}/\tau_0 = 1.25 \pm 0.15\) (n=5) in epifluorescence, in agreement with the discussed expected underestimation of the contact angle in epifluorescence measurements.

../_images/Fig_03-E.png

Figure 475: Increase of the tension ratio between the tension \(\tau(t)\) at time \(t\) and the initial one \(\tau_0\). Same color code for same doublets as in figure 472, 473 and 474. The actin cortex rupture in the blue dashed line also presents the highest relative tension increase.

Cortical tension increase in doublets and in cells

In cells, the cortical tension can be as low as 50 pN/µm in fibroblast progenitor cells [Krieg et al. 08] and can go up to 4000 pN/µm for dictyostelium [Schwarz et al. 00]. Surprisingly, when myosin activity is affected, either by drugs or by genetic manipulation, the cortical tension only decreases by a factor of about 2. Cells are also observed to round up during division, where a tension increase by a factor of two is sufficient [Stewart et al. 11], [Kunda et al. 08] . Our in vitro reconstruction is able to reproduce similar changes of cortical tension, as we observe a cortical tension increase by a factor of up to 2.4.

Different contributions for cortical tension

The cortical tension is the sum of the membrane tension and the tension due to the acto myosin cortex. We questioned how the membrane could contribute to the cortical tension and in our assay, we showed that in some cases, it might account for approximately 50% of the cortical tension. In suspended fibroblast cells, the membrane tension is estimated to amount to 10% of the cortical tension [Tinevez et al. 09]. When the actin polymerisation is stimulated, the cortical tension is multiplied by a factor of 5, also showing a strong dependence with actin dynamics [Tinevez et al. 09]. Therefore, the residual tension in cells might be due to actin dynamics, which is absent in our experiments. How actin contributes to cortical tension is still an open question that needs to be addressed to the cell geometry. Whereas it has been proved that actin polymerisation outside a liposome generated inward pressure, the way this observation can be translated into tension in a different geometry is not yet clear. In vitro assays are on their way to mimic actin dynamics in cells [AbuShah et al. 14]. They will allow to unveil the mechanisms of tension build-up by actin dynamics, the last remaining module that still needs to be understood, as the effect of both myosin and membrane have been clarified in this study.

Conclusion

We provided a biomimetic reconstitution of the tension build-up by acto-myosin contractility, through the use of liposome doublets. Cortical tension changes were visualised in situ over time by analysing doublet shape changes. This method allowed us to directly quantify the relative increase in tension due to myosin, regardless of the one due to actin dynamics. So, a thorough understanding of the composite systems contraction, rebuilt brick by brick to finally model a living cell, will hopefully lead the way towards a reconstitution of complex systems like tissues.

3D fitting

It remains challenging to obtain the doublets geometrical parameter, as in classical phase contrast and epifluorescence microscopy, the acquired images only capture a single focal plane of the doublets. This makes the analysis difficult, as the observation plane should be the equatorial plane of the doublet.

In order to achieve a good precision in the measurements of the contact angle, we decided to use confocal microscopy and to acquire evenly spaced z-stacks. The doublets 3D structure was reconstituted from these stacks, thus allowing to recover the geometrical parameters and the contact angle.

In order to determine the geometrical doublets parameters, we modelled them as two intersecting spheres, determined the expected 3D images and adjusted the model parameters to resemble the obtained experimental data.

I was responsible for developing a fast and precise method to reliably and automatically recover the liposome doublets geometrical parameters, based in the image stacks, acquired by using spinning disk microscopy. In the following part, I will develop the principle of this method and the result on liposomes doublets.

First step: Fitting a single liposome

In this part, we will describe the principle that allowed us to determine the 8 geometrical parameters that characterise a doublet: 2 centers (X,Y,Z) and 2 radii (\(R_1\) and \(R_2\)).

As the principle for finding the geometrical parameters does not differ with the number of dimensions, the presented methods can be applied even in higher dimensions (e.g. deformed ellipsoid liposome, or multi channel imaging). Furthermore, as the principles also remain similar in a space with less dimensions, we will restrict our discussion to a single liposome in a 2D plane (X,Y position of centers and R, radius, hence reducing the parameters to be determined to six instead of eight).

Experimentally, liposomes are observed using fluorescently labeled actin that forms an homogeneous micrometer-sized actin shell. In the observation plane, the liposome is a bright ring of a predetermined thickness (we will refer to this as the expected signal). We can observe the experimental noise on top of the image, where the principal sources are identified as the presence of fluorescent actin monomers in the buffer solution and electronic noise from the CCD camera. Eventually, the noise in the outside buffer, due to monomeric actin, can be higher than in the inside buffer, which is actin-free.

The signal from a liposome and the addition of noise can be replicated numerically as per figure 476.

liposome Model

Figure 476: Left : A simulation of liposome fluorescent image consisting of a uniform shell or membrane (expected signal). Middle: Same Image Adding Gaussian noise. This simulates one plane of a confocal Z-stack. Right: Liposome simulation with a fluorescently labeled actin shell in a fluorescent external buffer and a non fluorescent inside buffer.

The expected signal can be modelled numerically, using several parameters of the system (center and radius of liposome, point spread function of microscope, ...).

In order to find the correct parameters for the doublets, we will numerically correlate the acquired data with the numerical model and search for the correlation that best corresponds to the real image. The correlation between the model and the images data can be. expressed as :

(7)\[r_{xy}=\frac{\sum\limits_{i=1}^n (x_i-\bar{x})(y_i-\bar{y})}{(n-1) s_x s_y}\]

In which \(x_i\) are the luminosity values of each \(n\) pixel in the acquired data, \(y_i\) represent the pixels luminosity in the model \(\bar{x},\bar{y}\) correspond to the average values over the images, \(s_x\) and \(s_y\) are the standard deviation of the luminosity values.

As the monomeric fluorescently labeled actin and the electronic noise are dominant in the acquired images, we can assume a uniform noise on top of the expected signal. The correlation between the model and the noise is on average uniform. .. math:

:label: eqa403

r_{noise,model(params)} = cst

And the correlation between the expected signal and the model is expected to be maximal for the model parameters equivalent to the real geometrical doublets parameters.

(8)\[{arg\,max}_p\left(r_{data,model(p)}\right)= {arg\,max}_p \left(r_{expectedSignal,model(p)}\right)\]

In which \({arg\,max}_p\) stands for the argument of the maximum, that is to say, the set of points of the considered argument, for which the given function reaches its maximum value. Thus, searching for parameter values that maximise the correlation between the model and the data, implies that we find the geometrical parameters we are interested in.

We can test the ability to do this numerically by generating data, adding noise to it and trying to recover the parameters of the expected signal.

By looking at the correlation value between the generated data and the model as a function of model parameters, we can check that the correlation values are maximal when the model center value corresponds to the expected signal center value (Fig 477), and when the radius of the model liposome has the same radius in the model and corresponds to the radius in the generated data (Fig 478).

liposome Model

Figure 477: Correlation value as a function (arbitrary units) of two among the fit parameters. The liposome radius in the model is taken as equal to the value of the expected signal, and the position of the center is varied? in the X and Y directions. The correlation value is maximal for the center position in the model, that equals to the center of the expected signal. The local maxima observed on the 3D representation are well below the global maximum value. The peak at the global maxima is sharp, hinting that the search of the maxima requires relatively good initial parameters (lower than ~1/10 of the liposome radius). The sharpness of the peak point corresponding to the best fit parameters on experimental data, should be robust.

../_images/c-R-_100-by-100-RC-40_0-noise-0_5-delta-4_0_.png

Figure 478: Same as figure 477 with Y, center position taken as equal to the expected signal, variating X position of the model and radius of the liposome. The graph shows the same properties as before.

We could search the parameter space of the model and maximise the correlation between the model and the experimental data through the use of minimisation techniques. We then could recover the liposomes geometrical parameters by efficiently computing the correlation value within a few hundreds of points, thus giving access to the liposomes’ geometrical parameters, in this instance position and radius.

Fitting a doublet

The determination of the contact angle on epifluorescence images or phase contrast images often results in an underestimation, as the imaging plane is not necessarily one of the doublets equatorial planes. Moreover, most determinations of the contact angle on phase contrast and epifluorescence images are handmade [Maitre et al. 12] and are subject to experimenter’s bias, as the experimenter draws the tangent lines at the contact point between the liposomes. Thus, we decided to develop fitting routines for the acquired 3D confocal stacks. In our case, we avoided the usage of fluorescent lipids that could artificially change the membrane tension.

As sketched in figure 462, the doublets are covered with a thin micrometer-thick layer of fluorescent actin filaments, which we imaged by confocal spinning disk microscopy. As the actin-layer is attached to the membrane and the contact angle is defined as the angle between the lipid bilayer, imaging the actin-layer corresponded to the angle between the inner surfaces of the two actin networks present on each liposome.

Thus, in order to determine the geometrical parameters of the doublets we also needed to model the actin shell. As the liposomes in contact consist of two spherical caps, the uniform actin layer will also form two spherical caps with a given thickness. The total image is thus the union of two spherical caps blurred by the point spread function of the microscope. This can be seen on figure 479. We can notice on this image the presence of the doublet, lying on the chamber surface. We checked in this case that the contact surface between the chamber and the doublet did not change during experiments.

../_images/max_proj_340A.png

Figure 479: Maximum projection along X,Y and Z of recorded stacks, green channel represents actin. One can observe that the liposome doublets are lying on the surface of the observation chamber (arrows).

As the doublets contraction is rapid, and the recorded 3D stacks contain a large number of frames, it is hence crucial to be able to compute the model and the correlation in a reasonable time (less than an hour per images). To achieve this besides calculating the model as efficiently as possible, one can replace the exact calculation of two spherical caps and the point spread function of the microscope by the union and subtraction of pre-calculated spheres followed by a 3D numerical Gaussian blur (Fig 480).

../_images/3dblur.png

Figure 480: Principle of numerically approximating the two spherical caps as intersection of two spheres, followed by a 3D numerical Gaussian blur. Compared to the exact calculation of the fluorescent density, the numerical speed-up allows to make fits on doublets in minutes instead of hours.

However, the use of such numerical techniques is not devoid of artefacts. In the particular case of discreet Z-stacks that are not sufficiently spaced, the different radii in the fluorescent rings within subsequent stacks can lead to a “ring-artefact” (Fig 481), when using numerical Gaussian blur. In the case of a too pronounced “ring-artifact”, a “ghost” spheres can appear around each liposome, liable to cause the doublets fitting process to fall into a local maximum of correlation, thus leading to a wrong value of the geometrical parameters.

../_images/ring_artifact.png

Figure 481: Left : One plane of the numerical model with an exaggerated ring artifact due to an under sampling of the model in the Z-direction, stacks from “Far” Z leak onto the current Z-plane and forms a ring. Right : Same model plane with enough sampling plane in the Z-direction, does not show the ring artefact. In this case, we used a sampling with the same number of slices than the recorded data. (X,Y in arbitrary units)

In our case, we had a sufficient number of planes per stack so that
the numerical model with the same sample size similar to the data, did not show the ring artefact and had

a smooth transition near the position of the spherical cap. Though, the ring artefact could be eliminated by oversampling/interpolating the model before the numerical Gaussian blur and undersampling afterwards to reach the correct number of pixels.

The size of the Gaussian blur, which will act as a regularisation function for the value of the correlation between the model and the acquired data, can also be adjusted to be higher (see Fig 482), thus smoothing or eliminating local maxima, but reducing the precision in the maxima position.

../_images/max_proj_model.png

Figure 482: Maximum projection along the X,Y and Z of numerical model, the “ring” effect is still slightly visible near the pole of each liposome, but is not sufficient for the fit to be stuck in a local minimum.

The correlation value between the model and the experimentally recorded data can be maximised, by using already available functions, and more particularly the Nelder–Mead simplex algorithm as implemented in scipy.optimise python library. This provided us with the 8 parameters of the doublets. Result of the fits are shown in figure 483.

../_images/Doublet-402-A-Fit-t-0.png

Figure 483: Maximum projection of confocal images in the X,Y and Z projections as well as the result of the fits shown as equatorial circles for the three directions

of projection.

Using the fast Cython code ([Seljebotn 09]) also allowed to speed up fitting to a reasonable time: one Z-stack of 3 millions pixels can be fitted in about 40 seconds, thus allowing the fitting of a full 3D movie of a doublets contraction to be completed in less than an hour for 30 to 40 frames.

To ensure fits robustness to doublet center displacement during acquisition, the initial parameter of the fit were chosen manually for each first frame of each sequence. The final fit parameters of each frame are reused as initial fit parameters for the subsequent frame.

In order to test the robustness of the fit, we randomly modified the initial fit parameters by +/- 1µm, and we checked that the final parameters did not vary.

For a couple of parameters, the correlation function values can be plotted to check the regularity of the function and the absence of local maxima. Figure 484 and figure 485 show the resulting correlation values.

../_images/gof-2d-doublets.png

Figure 484: Correlation of the model and the data as a function of the center position of one of the model spherical caps along the X axis and the radius of this same spherical cap. Vertical axis in arbitrary unit.

../_images/gof-3d-doublets.png

Figure 485: 3D representation of the data in figure 484, the function shape is the same as the simulation done with the expected signal in figure 477 and 478

The fit correctness is also checked visually to prevent errors in the procedure. We found that the fit was systematically accurate and coherent with the manual measurements of the contact angle. Whenever the red channel was also present and the liposomes contained sulphorhodamin B, we were additionally able to visually check the fits, by using the maximum projection of the red channel. (see Fig 486).

../_images/srhod_superimpose.png

Figure 486: Maximum projection of the red channel (sulphorhodamin) and fitted parameter for the doublet.

Discussion

This part aimed at demonstrating that by modelling the liposome doublet and using fluorescently labeled actin, we were able to develop a technique that automatically and robustly determined the geometrical properties of the liposome doublets.

We noted that the red fluorescent dye present in the inside buffer of the liposome could be used conjointly for the green channel, in order to improve the fit quality, though this would require the extra parameters of the interface radius. As the required computation time to fit the doublets increases rapidly with the parameters number, we came to the conclusion that this solution was impractical. Moreover, the interface curvature is relatively small and the difference between the curved interface and a flat plane close to the optical resolution, hence the risk for the fits to become unstable. The use of fluorescently labeled lipids for the liposome membrane also suffers from the same issues of extra parameters, if one wants to recover the interface position.

Conclusion

We developed a robust and automated method to determine the geometrical parameters of liposome doublets. This allowed to robustly determine the liposome doublets geometrical parameters without any experimenter’s measurements bias, thanks to the selection of the illumination plane, the resolution of optics and the luminosity scale.

We determined that liposome doublets with reconstituted acto-myosin cortices were a biomimetic system allowing to measure the changes in cortical tension with time. 3D fitting helped to quantify the tension by obtaining the corresponding contact angles.

A simultaneous observation of the contraction of multiple liposomes doublets and the ability to automatically determine the geometrical parameters allowed the collection of more samples. A faster and more reliable data acquisition on actin network contractions will lead to a better understanding of the effect of actin network in vitro thus also paving the way for the reconstitution of more complex systems.

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