Predicting nucleation sites in chemotaxing Dictyostelium discoideum

Blebs, pressure driven protrusions of the plasma membrane, facilitate the movement of cell such as the soil amoeba Dictyostelium discoideum in a three dimensional environment. The goal of the article is to develop a means to predict nucleation sites. We accomplish this through an energy functional that includes the influence of cell membrane geometry (membrane curvature and tension), membrane-cortex linking protein lengths as well as local pressure differentials. We apply the resulting functional to the parameterized microscopy images of chemotaxing Dictyostelium cells. By restricting the functional to the cell boundary influenced by the cyclic AMP (cAMP) chemo-attractant (the cell anterior), we find that the next nucleation site ranks high in the top 10 energy values. More specifically, if we look only at the boundary segment defined by the extent of the expected bleb, then 96.8% of the highest energy sites identify the nucleation. Author summary This work concerns the prediction of nucleation sites in the soil amoeba-like Dictyostelium discoideum. We define a real valued functional combining input from cortex and membrane geometry such as membrane curvature and tension, cortex to membrane separation and local pressure differences. We show that the functional may be used to predict the location of bleb nucleation. In the region influenced by the cAMP gradient (the cell anterior), the next blebbing site lies in the ten highest energy functional values 70% of the time. The correctness increases to 96.8% provided we restrict attention to the segment in the general location of the next bleb. We verify these claims through the observation of microscopy images. The images are sequential at 1.66 and 0.8 seconds per image. We first identify the earliest sign of the bleb. We then use several observational factors to identify the nucleation site and estimate the corresponding location in the prior image.


Introduction
For Dictyostelium discoideum (D. discoideum) as well as various mammalian cell types, 14 the means of motility changes depending on varying conditions. At least since Yoshida 15 and Soldati [37], we have known that D. discoideum cells in three dimensional 16 environments often use blebs as a means of cell extension. These are pressure driven 17 blister-like protrusions of the cell membrane. Further, the greater the pressure from the 18 environment, the greater the reliance on blebs during migration. In humans, well-known 19 cases of cells using blebbing to migrate include embryonic organ development and 20 cancer cell migration ( [6], [33], [15], [26], [2], [13]).
there are no longer high energy spikes and the cost of maintaining the shape is lowered. 71 Secondly, we focus our attention to the area of the cell affected by the gradient of 72 cAMP, the area of blebbing activity. In our setting, this is roughly half the cell boundary. 73 Then we ask whether the energy functional can identify the specific nucleation site. 74 For chemotaxing and confined D. discoideum cells under 0.7% agarose, we identify 75 newly formed blebs and use various observational methods to identify the nucleation 76 site. These cells generally exhibit high energy profiles in the blebbing region. For cases 77 that we can infer the nucleation site in the image prior to the bleb, this location 78 coincides with the local high energy location identified by the functional in over 96% of 79 the observed cases. Furthermore, when applying the functional to the general region 80 affected by cAMP, our energy functional located the next nucleation site 40% of the 81 time as the first or second top energy value.This finding is comparable to results 82 reported in [9]. 83 This project has used differential geometric and local pressure differential to predict 84 bleb initiation site. In addition, in the process of locating nucleation sites, we have 85 noted frequent evidence of cortex degradation just prior to blebbing. This validates the 86 point of view that cortex ruptures initiates belb nucleation. We have tabulated these 87 events below. Subsequent biological research may be able to clarify the interplay 88 between the differential geometry and the biology in bleb site selection and explain the 89 observed early cortex disassembly. 90 This article is organized as follows. In the Methods and Materials section, we review 91 the cell preparation and microscopy technology. We state the geometric procedures underlying this work and the basic boundary energy functional to be used. for LifeAct-GFP expressing Ax2 wild-type cells [27]. Cells were starved for cAMP oil immersion objective). Data collected using both GFP and RITC channels resulted in 120 one frame per 1.66 seconds where data collected using only GFP resulted in one frame 121 per 0.800 seconds. ImageJ was used to adjust the brightness and contrast of the images, 122 which were then imported into our Mathematica based geometric system.

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Digitizing Microscopy Images 124 We use our own system to render photographic images as objects in Cartesian space.

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In [27], we detail the system components and provide reason that this is appropriate for 126 this setting. µm.) We refer to these points collectively as the EquList. This is the foundation for the 132 discrete form energy functional.

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An Energy Functional 134 We use a membrane energy functional related to the one introduced by T. Bretschneider 135 in [34] and [9]. This functional is a modified version of one used in a computer graphics 136 application [18], which in turn is modified from one that arose in modeling red blood 137 cells [14]. Our energy functional (1) is identical to the one defined in [9] save for the last 138 term in the integrand.

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In another setting, something similar was used to derive a model for blebbing [21].

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In this paper and [9], the energy functional is used to identify locations likely to bleb.

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In this section, we use a third version of the functional to compute energy values. Our 142 changes are discussed below. Our purpose is the same.

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Note that the fourth term of the integrand in (1) is not energy. Hence, the integrand 144 is not energy and the integral is not total energy. We denote it as pE total for 145 pseudo-energy, and refer to it as energy as the alternative is cumbersome. Moreover, 146 others have called a similar expression energy [9].
with the notation. Points on the membrane are denotedx = (x(s), y(s)), points on the 149 cortex byx c = (x c , y c ), while link lengths are inferred by the distance between the 150 curves. In the discrete model this reduces to the distance between corresponding points 151 in the respective point lists.

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The total energy of the complex is expressed as the point-wise energy integrated 153 (summed) along the cell boundary.
where Λ denotes the total boundary length. The individual terms are described as 156 follows.

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The first two terms refer to the membrane mechanics with membrane stiffness 158 parameter, α, and bending rigidity parameter, β. The term m 0 is the resting 159 membrane length. The first term is the energy derived from the Tension, the 160 second term is the energy required to bend the membrane [16]. Parameter rest will be about 0.04µm. The spring constant κ is resolved below.

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The last item is the hydrostatic pressure. It is not formatted as energy. The are canceled by other small changes in the cortex/membrane. But this process is 175 not perfectly efficient. Hence, we accommodate for small local variations in 176 pressure. In [4], this effect is identified in cancer cells. In [2], there is a similar 177 report for zebra fish cells.

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Since we cannot measure pressure directly, we settle for an alternative means. If 179 the actin cortex is contracted, we expect increased separation from the membrane. 180 As a result, the linking protein is stretched. Hence, we use changes in the linking 181 protein lengths as a replacement for pressure changes caused by Myosin II 182 contractions. 183 We caution the reader that the first term of the integrand of (1) is not a constant as 184 it might appear. Rather, the statement in (1) represents the energy at the initial values. 185 Upon initialization, the membrane seeks a position that minimizes the functional. As 186 relaxation occurs, the membrane parameterization is no longer arc length. See also [28]. 187 We may use the integrand in (1) to define pointwise or local energy.

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The Discrete Energy Functional

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In order to use the local functional (2) we needed to know the location of the membrane. 193 We knew the cortex as a cubic B-spline or via the EquList {x c i } of equally spaced cortex 194 points. Furthermore, we knew that the membrane is so close to the cortex that it is 195 nearly indistinguishable in a microscopy image. We began by defining the membrane via a list of pointsx i =x(s i ) where each point 197 has distance l i fromx c i along a ray initiating at an interior point of the cell (usually the 198 cell centroid). We identified this distance with the length of the linking protein joining 199 x i andx c i . The issue now was to determine the l i . 200 We proceeded with the assumption that the cortex and membrane are nearly 201 identical, or equivalently, the l i are very small. Consequently, we supposed that the list 202 {x i } is nearly equally spaced. Furthermore, as we did not know the curve representing 203 the membrane, we used finite differences, specifically central differences, for the 204 derivatives in (2). Next, we restated the local energy functional in discrete form.
where ∆s is the designated fixed distance between points on the membrane list. In turn, 209 we defined the discrete total energy. This is in effect the distance between points on the 210 cortex list.
a function of n variables, where n is the length of the EquList.
Resolving the l i It remained to determine the l i , equivalently thex i . When that was done, we knew 214 pE total and each pE(x i ).

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First we set the parameters for (3) as follows:  These values were found in [34]. 221 We resolved the n unknowns by arguing that the complex relaxes from an initial 222 state and seeks a stable state. At this state, the energy attains a local minimum, so that 223 any small variation in any l i increases the energy. 224 We took an initial value l i = 0.03µM, the minimal distance between the membrane 225 and cortex. We proceeded with minimization via a gradient directed search [20] and 226 arrived at a local minimum for pE total . We accepted the l i andx i for this state. This  The third feature as in the old cortex scar in Fig 1B1. We observed a distinct narrow 258 gap. We have denoted this feature d. This is likely the means for the cytoplasm to fill 259 the nascent bleb with actin and other proteins sufficient to form a new cortex. This indicate which is primary.

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It may be impossible to identify a nucleation site. The most common problem was 273 we did not have an image of the bleb at the right time.

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In final analysis, for the example all indicators point to d as the nucleation site.

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Note that this conclusion is based solely on observation. Moreover, the example cell is 276 unusual as it is rare to have all these features present for a given bleb.

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When the nucleation point was identified, then the next step was to identify the 278 corresponding point on the prior image. Eventually, we applied the energy functional to 279 this location. Even though the cell in general was usually little changed from frame to 280 frame, this may not be true of the region near the bleb.

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The first step in this process was to locate points corresponding to the bleb shoulder 282 points. For the current example, we used the cortex features to identify points in the 283 prior state that corresponds to a and b. We have identified these as a and b in Fig1C. 284 Now, it is clear that the predecessor to d is the area we have marked as d .

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There is already a visible gap at d . This is not a rare event. In fact we have 286 observed it in over 70% of the blebs we have looked at. Refer to Table 1 line 5.

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An alternate approach was to look at the internal actin structures and GFP bundles.  those cases seem to be reasonable. 308 We have proved a correlation between bleb site selection and edge geometry. The 309 correlation is more conclusive than in prior works. In the next subsection we include 310 cAMP in the discussion on bleb nucleation. with small dots. The first observation is that the nucleation site we have been studying 316 is not included. This is the one point we know will nucleate at the next frame. Next, many of the high energy locations are at locations where the cell is convex. Bassed on 318 our previous studiy, these are places that are not likely to bleb [27].

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We considered several alternatives to the energy function. Each time we asked the same 341 question, does the alternative perform as well when asked to predict the next nucleation 342 site given the bleb shoulder points. This is the same question asked in the section 343 Energy and Bleb Nucleation. We looked at a total of 23 cells and 86 blebs.

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First we considered the energy functional before the minimization process. This 345 functional was only successful in 74 cases or 86% of the time. We concluded that the 346 unminimized energy functional defined by fixed length linking proteins is not effective. 347 Next, we isolated each term in the energy functional. In particular, we removed each 348 term one at a time. The result of this study is displayed in Table 2. isolate the anterior region will improve global prediction.

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There are three elements of this study that separate us from [9], the pressure term in 378 our energy functional is not constant, the underlying geometric platform is based on a 379 B-spline defined from the points returned by the edge detect and we have asked 380 different questions. The result of these changes is a highly successful predictive 381 functional that we believe will point to the origin of blebbing. Furthermore, Π = 0 is 382 equivalent to a constant pressure term. The data in Table 2 show clearly that this is not 383 a viable option. Hence, this is a critical extension of its predecessor. 384 We expect that while the energy framework is highly predictive, it still does not 385 account for all the complex processes influencing the selection of nucleation sites. Aided 386 by our mathematical model, biological experiments can now be designed to study the 387 local mechanism responsible for bleb initiation.

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In turn, more biology should be implemented into the energy functional. For 389 instance, Collier et al [9] report the uneven distribution of Talin at the posterior face.

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To include this information into the energy functional requires developing the functional 391 without equally spaced points, hence without the finite difference method. This is 392 beyond the scope of the current article yet within known numerical processes. This is a 393 matter looking at the several alternatives and determining what adapts best to the case 394