Simulation of regulatory strategies in a morphogen based model of ...

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International Conference on Computational Science, ICCS 2017, 12-14 June 2017, Zurich, Switzerland

Simulation of regulatory strategies in a morphogen based Simulation of regulatory strategies in a morphogen based of Arabidopsis Simulation model of regulatory strategiesleaf in agrowth. morphogen based model of Arabidopsis leaf growth. 3 Elise Kuylen1 ,model Gerrit T.S. Beemster2 , Jan Broeckhove , and Dirk De Vos4 of Arabidopsis leaf growth. 1 2 3 4 Elise Kuylen , Gerrit T.S. Beemster , Jan Broeckhove , and Dirk De Vos 1 University of Antwerp, 2 Antwerpen, Belgium Elise Kuylen1 , Gerrit T.S. Beemster , Jan Broeckhove3 , and Dirk De Vos4 1 [email protected] University of Antwerp, Antwerpen, Belgium 2 1 2 3 2 3 4 3 4 4

University of Antwerp, Antwerpen, Belgium [email protected] University of Antwerp, Antwerpen, Belgium [email protected] University of Antwerp, Antwerpen, Belgium [email protected] University of Antwerp, Antwerpen, Belgium [email protected] University of Antwerp, Antwerpen, Belgium [email protected] University of Antwerp, Antwerpen, Belgium [email protected] University of Antwerp, Antwerpen, Belgium [email protected] University of Antwerp, Antwerpen, Belgium [email protected] University of Antwerp, Antwerpen, Belgium [email protected] [email protected] University of Antwerp, Antwerpen, Belgium [email protected]

Abstract Simulation has become an important tool for studying plant physiology. An important aspect Abstract of this is discovering theanprocesses that leaf growth at a cellular To this end, Simulation has become important toolinfluence for studying plant physiology. Anlevel. important aspect Abstract we have extended an existing, morphogen-based model for the growth of Arabidopsis leaves. of this is discovering the processes that influence leaf growth at a cellular level. To this end, Simulation has become an important tool for studying plant physiology. An important aspect We have fitted parameters to match important leaf growth properties reported in experimental we have extended an existing, morphogen-based model for the growth of Arabidopsis leaves. of this is discovering the processes that influence leaf growth at a cellular level. To this end, data. A extended sensitivity performed, allowed us to estimate theineffect ofleaves. these We have fitted parameters to was match importantwhich leafmodel growth reported experimental we have ananalysis existing, morphogen-based forproperties the growth of Arabidopsis different parameters on leaf growth, and identify viable strategies for increasing leaf size. data. A sensitivity analysis was performed, which allowed us to estimate the effect of these We have fitted parameters to match important leaf growth properties reported in experimental different parameters on leaf growth, and identify viable strategies for increasing leaf size. data. A sensitivity analysis was performed, which allowed to estimate the effect of these Keywords: modelling and simulation, computational biology, leaf us development

© 2017 The Authors. Published by Elsevier B.V. different on leaf and identifyofbiology, viable strategies for increasing leaf size.Science Peer-review under responsibility of growth, the scientific committee the International Conference on Computational Keywords:parameters modelling and simulation, computational leaf development Keywords: modelling and simulation, computational biology, leaf development

1 Introduction 1 Introduction Mathematical modelling and simulation is becoming increasingly important in the bio-sciences, 1 Introduction particularly in the field of plant physiology. An important factor in this evolution from a

Mathematical modelling and simulation is becoming increasingly important in the bio-sciences, reductionist towards a systemic approach, theAn rapidly increasing knowledge on molecular particularly inmodelling the field of plant physiology. important factor in thisinevolution from a Mathematical and simulation is is becoming increasingly important thethe bio-sciences, processes that determine the form and function of plants. Large-scale (‘omics’) studies at thea reductionist towards a systemic approach, is the rapidly increasing knowledge on the molecular particularly in the field of plant physiology. An important factor in this evolution from level of DNA, RNA and proteins, as well as more focused studies, have deciphered more processes that determine the form and function of plants. Large-scale (‘omics’) studies at and the reductionist towards a systemic approach, is the rapidly increasing knowledge on the molecular more regulatory networks and pathways that govern cellular behaviour. level of DNA, RNA and proteins, as well as more focused studies, have deciphered more and processes that determine the form and function of plants. Large-scale (‘omics’) studies at the all cells in aand plant arepathways both biochemically and mechanically connected in amore so-called more regulatory networks and that governfocused cellular behaviour. levelSince of DNA, RNA proteins, as well as more studies, have deciphered and symplastic framework, it is important for modelling studies to include spatial dimensions. To Since all cells in a plant are both biochemically and mechanically connected in a so-called more regulatory networks and pathways that govern cellular behaviour. describe such a cellular grid the plant tissue in which cells are fixed with respect to their symplastic framework, it is important for modelling studies to include spatial dimensions. To Since all cells in a plant are both biochemically and mechanically connected in a so-called neighbours, several plant modelling platforms have been developed. Some of these are primarily describe such a cellular grid the plant tissue in which cells are fixed with respect to their symplastic framework, it is important for modelling studies to include spatial dimensions. To aimed at such theseveral simulation of plant and interaction plants with their environneighbours, plantgrid modelling platforms been developed. of these are primarily describe a cellular - thearchitecture plant tissuehave - inthe which cells areofSome fixed with respect to their ment [18, 8, 17], whereas others focus on the cellular level and the tissue level [6, 16]. The aimed at the simulation of plant architecture and the interaction of plants with their environneighbours, several plant modelling platforms have been developed. Some of these are primarily ment [18, 8, 17], whereas others focus on the cellular level and the tissue level [6, 16]. The aimed at the simulation of plant architecture and the interaction of plants with their environment [18, 8, 17], whereas others focus on the cellular level and the tissue level [6, 16]. The1 1

1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the International Conference on Computational Science 10.1016/j.procs.2017.05.238

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latter are vertex-based simulators that represent plant tissue as polygons of vertices that are connected by edges. These edges represent cell wall segments and have mechanical attributes. This makes these models highly suitable for simulating tissue and organ growth. Functioning as its central power-station, leaves are the basis for growth of the plant. Improving our understanding of leaf growth and development can potentially lead to crops with higher productivity and resistance to climate change [11]. While many experimental approaches exist for studying leaf growth regulation from a molecular perspective, we should be able to quantify leaf growth in a precise way. Various methods exist for this purpose, depending on the plant species being studied. Quantifying the growth of the experimentally widely studied plant Arabidopsis thaliana typically involves measuring the evolution of the leaf area, estimating the total cell number and calculating a number of derived quantities. This has led to the distinction of different phases during leaf development. After the start the leaf (primordium) primarily grows by uniform cell division. A division front and afterwards an expansion-only front gradually develop. Over time, both fronts move to the leaf base. Eventually, the growth zones disappear from the leaf, and after cell maturation leaf growth is finished [7]. Mathematical models have been proposed for several aspects of leaf development such as leaf initiation [3, 19], shape [14], serration [2], and venation [20]. Previously, we proposed a computational model which reproduces the main stages of leaf growth based on a single mobile signal or morphogen that is produced at the base [5]. In this paper, we present a leaf growth model, which we improved using experimental data from Kheibarshekan Asl et al. [13]. These data precisely quantify the evolution of cell numbers and sizes over time. Through a sensitivity analysis the influence of perturbation of model parameters and their associated processes on leaf area, total cell number and average cell area is investigated.

2 2.1

Methods Simulator framework

We have used a vertex-based simulator that has been developed in C++11, building on the VirtualLeaf framework [16]. The cells are modelled as polygons of nodes connected by edges, which represent the cell walls and membranes separating the cells in the tissue. Both cells and walls have attributes such as chemical concentrations or mechanical properties. Dedicated classes are used to implement the rules and equations that govern the dynamics of those attributes. The simulator moves forward in discrete timesteps, the length of which can be specified through an input file. This input file also contains the initial conditions of the cellular mesh as well as all model parameters. During each timestep, two important processes are executed. First, biochemical processes within and between cells are simulated. This includes intracellular reactions, transport processes between cells, evaluation of rules for cell division and so forth, which are all described in model-specific classes. Subsequently, the mechanical movements of the tissue are simulated with displacements of all nodes in a Metropolis-Monte Carlo based energy minimization algorithm. Leaf cells experience an internal (turgor) pressure that forces them to expand against the elastic forces from neighbouring walls. To find an equilibrium, we use a Metropolis algorithm (see [16] for details) to minimize an energy function (Hamiltonian) of the cell mesh. This Hamiltonian function can be expressed as follows: H = λA

  a(i) − Aτ (i) 2 i

2

a(i)

+ λM

 j

(l(j) − Lτ (j))

2



Simulation of a morphogen based model of et Arabidopsis Kuylen, Broeckhove & De Vos Elise Kuylen al. / Procedialeaf. Computer Science 108CBeemster, (2017) 139–148

The first term sums over all cells, and expresses the turgor pressure potential of the cells in the mesh. The resistance of cells to compression and expansion is represented by λA . The Aτ (i) are the target areas of the cells, while a(i) are the actual areas. The second term, which sums over all edges, represents the elastic potential energy of the mesh. The λM is the spring constant, the Lτ (j) are the target lengths of the wall segments represented by the edges, and l(j) the actual lengths of these wall segments. The algorithm loops over the nodes in random order, moves each node with a random displacement and accepts this move with an energy (and temperature) dependent probability. To reproduce the elongation of the leaf shape, we have introduced a bias (-3 µm) in the distribution of the displacements.

2.1.1

Leaf Growth Model

The initial tissue structure of the model has 32 cells and represents the Arabidopsis first real leaf (pair) at 4 days after sowing. The upper 16 cells of this mesh represent the actual leaf blade, while the lower 16 cells represent the petiole. The latter cells are fixed in size, shape and position. To reduce computation time, we have chosen to let one simulation cell represent 8 cells in reality. One length unit in the simulator is equivalent to 1.5 µm. In our model the characteristics of leaf growth are governed by a single mobile signal or morphogen that is produced at the base. For a given amount of time, it is generated there at a constant rate. Degradation of the morphogen happens continuously, in proportion to its concentration. From where it is produced at the base of the leaf, the morphogen is transported, cell-to-cell, throughout the rest of the leaf. The change of the number of molecules of the morphogen in a cell i over time can then be expressed by the following equation:   dMi ∆cji = + P − k d Mi lij D dt ∆x j The transport process is represented by summing over the j cells neighbouring cell i, with representing the diffusive flux from cell j to cell i. This happens according to Fick’s law, where ∆cji is the difference in morphogen concentration between cell j and cell i, and ∆x is the thickness of the wall segment between cell j and cell i. Furthermore, this flux is multiplied by D, a diffusion coefficient and lij , the length of the wall segment between cells i and j. Since edges in the model consist of both cell membranes and cell walls it is more accurate to use an apparent diffusion coefficient Dapp = D/∆x. As such the equation represents first order intercellular passive transport processes. An underlying assumption here is that the intra-cellular morphogen gradient can be neglected. P represents the cellular morphogen production rate, which is zero for cells belonging to the leaf blade. Finally, kd is the morphogen degradation rate. The concentration of morphogen in a cell determines both the growth and division behavior of the cell. To be able to divide, the morphogen concentration must exceed a certain threshold and a certain time must have passed since the cell last divided. Expansion of the cell is driven by the size of its target area. This target area increases according to a relative growth rate, when a certain threshold for the morphogen concentration is exceeded. This threshold is typically lower than the threshold for division. ∆cji ∆x

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2.2

Simulations

Results presented in this paper were obtained with an Intel(R) Xeon(R) CPU E5-2697 v3 2.60GHz CPU. Running the simulator for 150 timesteps, starting with a mesh of 32 cells, typically takes about 30 minutes on an otherwise idle system. To estimate optimal parameters for the leaf model, we fitted the leaf size, cell number and average cell area in function of time to experimental data. In addition, we ensured that the relative differences between target and actual cell areas in the leaf did not diverge. We also ensured that varying the random number generator seed had no significant effect on the fitted properties. Plots for various properties of the leaf were obtained using python scripts that used HDF5 files (support.hdfgroup.org/HDF5) as input. For analysis of the tissue files, we used the PyPTS (pypi.python.org/pypi/PyPTS) and numpy (www.numpy.org) libraries, while for the plotting itself we used the matplotlib (www.matplotlib.org) library. Figures of the simulated mesh were obtained using Paraview (www.paraview.org), with a plugin specifically written to read HDF5 plant tissue files. The project is portable over all platforms with a GNU compiler and the Boost (www.boost.org) library.

3 3.1

Results Model Development

Our previously published model [5], captured the overall features of Arabidopsis leaf growth [1]. In this work, we have extended and refined the model in accordance with experimental findings reported by Kheibarshekan Asl et al. [13]. Based on their data of leaf epidermal cells in Arabidopsis, they found that cell cycle duration is relatively constant throughout the leaf development and found no indications for a size threshold for division. We have therefore replaced the size based cell division rule in the model by the condition that cell division is possible after a minimum duration since the last division, briefly called a ’timer’. A second condition for division is that a specific morphogen concentration threshold is exceeded. This is important to spatially define a zone in which cell division is possible in the growing leaf blade. To avoid aberrantly small cells we also introduced a cell size threshold for division. This however, does not influence the overall growth of the leaf (results not shown). Due to the inherent (biological and experimental) variability of leaf growth data we have opted to manually fit our model by varying the model parameters and plotting the results against the three principal leaf growth curves from [13] (Figs. 1A,1B,1D). Those plots describe total leaf area, number of cells, and average cell area of the first real leaf (pair) as a function of time after sowing. The fitted parameter values are listed in Table 1, and the fit is presented in Figure 1, with plots for leaf area, number of cells, and the average cell area in function of the time shown in figures A, B and C respectively. Plots are also shown for experimental data from [13], [15] and [4], to give an idea of the variability of such data. We have set the timer for cell division at 22 hours, as suggested by the findings of [13]. Compared to the original model of [5], the morphogen production rate, the morphogen production duration, the morphogen threshold for division and for expansion, and the relative growth rate were increased, while the apparent morphogen diffusion constant was decreased. All three curves match those of the results presented in [13]. Evolution the total leaf area and the number of cells exhibits a similar slope as the experimental data. For the average cell areas, however, the slope could not be matched completely: the experimental data suggests a sharp increase in average cell area around day 11, while the simulation data shows a more gradual increase. 4



Simulation of a morphogen based model of et Arabidopsis Kuylen, Broeckhove & De Vos Elise Kuylen al. / Procedialeaf. Computer Science 108CBeemster, (2017) 139–148

Parameter name morphogen production rate morphogen production duration morphogen degradation rate constant apparent morphogen diffusion constant time threshold for division morphogen threshold for division morphogen threshold for expansion relative cellular growth rate

Symbol P n.a. kd Dapp n.a. n.a. n.a. n.a.

Original value 156AU/h 120h 0.01h−1 6µm/h n.a. 0.40AU/µm2 0.02AU/µm2 0.025h−1

Fitted value 3000AU/h 133h 0.01h−1 4.5µm/h 22h 4.44AU/µm2 0.133AU/µm2 0.038h−1

Table 1: Overview of parameters fitted for the leaf model, compared to the values of [5]. AU are Artificial Units for number of morphogen molecules) Initially, cell numbers and average cell area exhibited a stepwise evolution at the beginning of the simulation. All cells divided in a nearly syncronous way. This was an artefact of the initial tissue simulation input. The ‘time since division’ attribute of the cells is unknown and was set to zero for all cells. We therefore have randomized the ‘time since division’ attributes in the initial input, using a truncated normal distribution (between zero and the time threshold for division) with a standard deviation of 50% of the time threshold for division. Furthermore we added noise to the division duration at each simulation time step, using a random uniform distibution with a width equal to 10% of the original value of the timer. This yields an acceptable result with the curves for the number of cells and the average cell area both significantly smoothed out. We have also looked at the geometrical evolution of the leaf, and the evolution of the morphogen flow throughout the leaf. In figure 2 the situation 4, 8, 11, 17 and 29 days after sowing can be seen. To obtain a more elongated leaf shape, we introduced an upwards bias for node movements in the Monte Carlo algorithm.

3.2

Model Sensitivity Analysis

We performed a parameter sensitivity analyses to gauge the effect of the parameter perturbations. We have looked at 3 important output variables: total leaf area, number of cells and average cell area. We opted to perturb the principal model parameters locally each plus and minus 5% to obtain a significant response. We approximated the sensitivity to perturbation of a parameter as:    X+5% − X−5% Pref S ≈ S5% = Xref P+5% − P−5% The following parameters were perturbed: morphogen degradation rate, morphogen production rate, morphogen production duration, morphogen diffusion constant, morphogen threshold for both division and expansion, relative cellular growth rate and the time threshold for division. An overview of the results, is represented as heatmaps in figure 3. 3.2.1

Total Leaf Area

Increasing the morphogen degradation rate has a gradually increasing and strong negative effect on the total leaf area. Perturbing the morphogen production rate produces an opposite, albeit weaker, effect. Increasing the morphogen diffusion constant increases the leaf size at 5

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Figure 1: Plots for total leaf area (A), number of cells (B) and average cell area (C) in function of the amount of time passed since sowing.

the beginning of the simulation, but the effect wears off later on. This is likely caused by the spreading out of the morphogen, which shifts developmental fronts, but makes them less robust to changes in the morphogen supply. Increasing the duration of morphogen production, extends all growth phases, which manifests itself fully late in the development of the leaf. Increasing the morphogen threshold for division has a slightly negative effect on the leaf size. This may be due to a lower cell production, which can affect the propagation of the morphogen throughout the leaf. On the other hand, this might also be caused by the resulting mechanical changes of the tissue. As expected, increasing the morphogen threshold for expansion negatively impacts the overall size of the leaf. The effect is strong, and manifests itself in the later stages of leaf development. An increase of the relative cellular growth rate increases the leaf area significantly, but the effect is attenuated, probably because the developmental fronts are not too strongly affected. This underlines the robustness of such a regulatory mechanism to changes in cell growth. Increasing the time threshold for division has a similar effect to increasing the morphogen threshold, but the effect is much more pronounced. 6



Elise Kuylen et al. / Procedia Computer Science 108C (2017) 139–148 Simulation of a morphogen based model of Arabidopsis leaf. Kuylen, Beemster, Broeckhove & De Vos

Figure 2: Leaf simulation 4 (A), 8 (B), 11 (C), 17 (D) and 29 (E) days after sowing. Figures were resized for better visibility. Colouring depends on morphogen concentration (yellow for high levels, red for lower levels, white for concentrations below 0.133 AU/µm2 ). 3.2.2

Number of Cells

Increasing the morphogen degradation rate has a negative effect on the cell number: less pronounced than the effect it had on the leaf area, but earlier. Perturbing the morphogen production rate, has an opposite, but weaker, effect. The effects of perturbing the morphogen diffusion constant can be seen very early in the simulation. However, the effect quickly diminishes as the simulation progresses, as was the case for the effect on the total leaf area. The magnitude of the influence is also similar to that on the total leaf area. The effect of perturbing the morphogen production duration on the number of cells is strong and early, but not quite as strong as the effect on the leaf area. In general, cell number curves converge earlier, and effects will typically manifest early too. Increasing the morphogen threshold for division has a strong negative progressive effect on the number of cells in the leaf: the division front is simply shifted basally. The effect on cell numbers of perturbing the morphogen threshold for expansion is very small, even less than that of perturbing the threshold for division is on the leaf area, and negative. This is possibly because of an earlier cessation of the leaf growth, which inhibits cell proliferation. An increase in relative growth quickly and dramatically decreases cell numbers, whereas it positively affects leaf area. Due to their faster growth, fewer (and bigger) cells are in the division zone. The cells divide with the same timer mechanism but they move past the division front faster which gives them less time to divide multiple times. Increasing the time threshold for division has a strong negative effect on the number of cells in the leaf - as can be expected, since it is the most important factor in determining whether a cell will split. 3.2.3

Average Cell Area

Most of the results can easily be derived from the above trends but the most remarkable observations are worth pointing out. The negative effect of increasing the morphogen degradation rate is initially more pronounced on cell numbers than on leaf area. Hence the relative increase in cell area is followed by a decrase as soon as the leaf area effect catches up. Increasing the morphogen production rate again has the opposite effect. However, the effects balance out completely in time: the leaf grows larger but the cell proliferation compensates it. Increasing the morphogen diffusion constant causes an initial fast increase in cell numbers which produces on average smaller cells, but after a short time this effect fades out. This corresponds to a more sudden transition from uniform cell division to the mature form - i.e. a shorter expansion 7

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phase. The effect of increasing the duration of morphogen production is somewhat similar to that of increasing the production rate, but later: the leaf becomes larger, with bigger cells. The effects of perturbing morphogen thresholds for division and expansion are in accordance with the dominant effect on division and expansion respectively. The effect on the cell size of perturbing relative growth is very strong (exponential). This suggests that varying the morphogen thresholds is not an ideal way to increase the leaf size if one is bound to a timer-like cell cycle mechanism - although it is the fastest mechanism. Increasing the time threshold for division has a strong positive effect on the average cell size. However, this does not mitigate the decrease in cell numbers: the total leaf area still shrinks when increasing the time threshold for division - as remarked above.

Figure 3: Heatmaps representing sensitivity of leaf area (A), cell number (B) and average cell area (C) to perturbations in specified parameters, in function of the time.

4

Discussion

Several computational modelling studies on leaf development exist (see the review in [11]). The model presented here is specifically aimed at quantitative growth characteristics of the dicot leaf. It is not intended to capture detailed changes in leaf shape as in [2, 14] or cell shape [9]. Leaf growth of our experimental type plant Arabidopsis thaliana was represented in two spatial dimensions. We focused on modelling the superficial cell layers, thereby neglecting that multiple layers are present in a vertical cross-section [12]. Our approach concurs with the experimental data that are reported and is supported by the assumption that surface layer dynamics are driving leaf growth. Some aspects of the overall leaf blade shape are not reproduced by our model and indicate that more work is required on the underlying mechanics. In future models anisotropic growth will possibly be modelled by more morphogens that define different principal growth axes [14]. Such efforts would also require a better understanding of the complex regulatory circuitry involved [11]. A sensitivity analysis was done with single local parameter perturbations. The changes in leaf area, total cell number and average cell area as a function of developmental time were analyzed and revealed some interesting findings. We discuss them here as if the perturbations emulate regulation of elementary processes based on underlying molecular interactions. Most parameters had an impact on one or more output variables. However, most perturbations affect both leaf area and cell numbers. The exceptions are the threshold values for cell division and expansion. They represent the spatial (non-cell-autonomous) boundaries of division and 8



Simulation of a morphogen based model of et Arabidopsis Kuylen, Broeckhove & De Vos Elise Kuylen al. / Procedialeaf. Computer Science 108CBeemster, (2017) 139–148

expansion zones and are suitable to tune cell numbers and leaf area, respectively. However, this does not conserve cell area and having a too small or large cell area can be harmful or even lethal to cells [21]. Quantitatively the strongest (and earliest) effects follow when the cellular growth rate or division rate are altered. Directly changing cell processes works multiplicatively. A possible downside is that tuning these strongly affects cell area, assuming that there are no compensating changes in other parameters. From a biological perspective a decreased cell division rate could lead to weaker cell growth. However, we observed a pattern of opposite sensitivities in cell numbers and cell areas in our heat maps. This is reminiscent of so called ‘compensation’, or enhanced cell expansion associated with a decrease in cell number during leaf development [10]. A biologically robust leaf size regulation may rather be associated with a strong and temporally consistent effect on leaf size by tuning one process, while conserving the leaf area. Based on our analysis, tuning of the morphogen production rate in the petiole cells or changing the supply rate from other parts of the plant would be an optimal strategy. Whereas these findings on biological strategies based on a mathematical model are interesting, further investigation is required. In particular, comparative analysis of models that have different basic assumptions regarding the interdependence of cellular growth regulatory processes is necessary. An important condition would be that alternative models are able to reproduce the same quantitative growth data. Further refinement of our leaf growth model is also important, with a focus on cell wall mechanics and molecular details. From a computational point of view this will require faster simulations at full cellular resolution.

Acknowledgements This work was supported by a grant of the Interuniversity Attraction Poles (IUAPVII/29:Growth and Development of Higher Plants) of the Belgian Federal Science Policy Office (BELSPO).

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