To test this model in a concrete molecular context and confirm the assumptions A that underline the usage of deterministic, isotropic, and distance-dependent forces in the interacting particle model Sec Force-balance and force-screening , we turned to a detailed agent-based simulation of microtubule-generated mechanics of the multi-nuclear cell.
Such a model allowed examination of whether stochastic effects are negligible, and whether MT bending and resulting elastic forces result in unforeseen effects. We choose to use the microscopic, stochastic simulation tool Cytosim [ 24 , 33 ], which has been successfully applied to a wide range of cell biological problems [ 34 — 36 ].
Using Cytosim, we simulated hundreds of MTs per nucleus, which are distributed uniformly around the nuclear circumference, cantilevered in the nuclear envelope and grow in a radially symmetric way. Individual MTs are treated as elastic rods, with the length of each MT characterized by the stochastic dynamic instability process [ 29 ], whereby each MT undergoes repeated stochastic cycles of growth, catastrophe, shortening and rescue.
Contact of a growing MT end with a neighboring nucleus or cell boundary results in MT bending, which generates an elastic pushing force. While pushing forces do not decrease the growth rate of MTs, catastrophy rates can increase to a maximum of 0. Similar force events would be observed in stochastic simulations of forces exerted by a kinesin plus-end motor. Thus we did not include explicit molecular descriptions of kinesins in these simulations. The sum of pushing forces from all MTs constitutes the net force.
We used the microscopic model, first, to simulate the interactions of a nucleus with a cell boundary in isolation. We placed a nucleus at a set distance from the single cell boundary Fig 7A left , allowed the MT dynamics to establish for a time period, and then released the nucleus, which started to move away from the boundary.
Repeating this experiment for different distances, we used the resulting nuclear speeds to calculated the net force as a function of nuclear distance Fig 7B.
To confirm that this procedure yields correct values, we compared the obtained forces to analytical approximations for the case of a single pushing filament. We found very good agreement between both methods see S1 Text for details. Second, we simulated the interactions of two nuclei in isolation. We placed nuclei at different distances from to each, let the MTs equilibrate and released the nuclei Fig 7A right. From the speed of the nuclear divergence, we computed an effective distant-dependent force Fig 7B.
We found that the forces in both cases rapidly decrease with distance for details see Methods. It should be noted, however, that also more complicated, non-monotonic force-distance relationships have been observed [ 37 ], which can affect the equilibrium positions of the nuclei. The precise behavior will depend on the MT dynamics and will have to be revisited when more details about muscle cell MTs become available.
A: To study the force acting on a nucleus created by pushing microtubules, we placed a nucleus near a boundary left and near a second nucleus right and used the speed to calculate the force details in Methods. B: Force calculation results from 5 realizations at 7 distances of A dots and power law fits of their average. C and D: Comparison between the microscopic simulation using Cytosim and the trajectories of positioning nuclei obtained from solving Eq 1 using the parameters obtained from B.
Movies comparing the simulations are shown in S1 and S2 Video. Parameters of the stochastic agent-based simulation can be found in Table 2 , those of the interacting particle simulation are given by the fit in B.
In a multinucleated cell, the assumption of pair-wise additive forces could be violated through a variety of factors. For example, if three nuclei are located along a straight line, the nucleus in the middle will, most likely, block direct interactions between the two outer nuclei shielding effect. To test the importance of possible non-pair-wise interactions and the validity of the particle interaction model, we simulated the positioning mechanism in a typical VL3 and VL4 cell.
We used random initial positions and actual cell widths, but only half the typical cell length and number of nuclei. This maintained the ratio of cell length and number of nuclei observed in the experimental data, while saving computational time.
Fig 7C and 7D shows that the simulated nuclear patterns are as observed in the experiment and predicted by the simple particle interaction model using pairwise forces, with single and zigzagged double-file spanning the cell length in narrow and wide cells, respectively.
Moreover, the trajectories of individual nuclei from their initial positions to the final, equilibrium positions were strikingly similar in the microscopic agent-based model and the particle interaction model Fig 7C and 7D. For the latter, we used the distance-dependent forces calculated from the stochastic nucleus-boundary and nucleus-nucleus simulations to inform the values of the force exponents and force magnitudes in Eq 1.
The striking similarity between the stochastic and deterministic trajectory validated the simple particle interaction model. Further these results suggest that, due to the rapid decrease of the net force with distance, non-pairwise interactions and possible shielding effects are negligible.
The idea is to start with multiple possible models and to use the predicted data to eliminate as many models as possible, dependent on their ability to recapitulate biological systems. This approach was successfully applied to cell signaling dynamics, metabolic networks, cell cycle, and spindle geometry [ 28 , 39 — 41 ]. In some instances, a small number of models can be analyzed one by one, as in a recent study on chemotaxis model inference [ 42 ].
In other instances, the number of model variants is so great that an unsupervised or semi-supervised computer screen of the models is necessary [ 28 ]. We searched computationally for the types of forces that could occur between pairs of nuclei and between nuclei and the cell boundary and could lead to positioning of the myonuclei.
A similar problem, mitotic spindle positioning, has a long history [ 21 ] and reductionist modeling proved helpful in that case. However, an approach philosophically similar to ours was recently applied successfully to search for forces positioning the sperm MT aster in sea urchin eggs [ 43 ]. We started with a large number of potential forces and formulated a few hundred potential models, each characterized by a few mechanical parameters.
We then used 1. We filtered out the vast majority of the models that were not able to predict the uniform spread of the nuclei along the cell long axis or the tendency of the nuclei to self-organize into the single file in narrow cells and double file in wide cells. These tests left us with two possible models, the parameters of which were fully determined by requiring the models to quantitatively fit the data in imaged cells.
The remaining two models made three non-trivial predictions: 1 the double-file pattern in wide cells is a zig-zag; 2 the average nuclear position along the cell short axis has the forked bifurcated dependence on the cell width, and 3 nuclear density is higher near the cell poles.
Remarkably, these two models make opposite predictions about the nuclear shapes. One of the models predicts that the nuclei have ellipsoidal shapes with the long axes oriented perpendicular to the cell long axis, which is contrary to the experimental data. Incidentally, this model is also less robust than the other, ultimate, model, which not only predicts correctly that the ellipsoidal nuclei have long axes oriented along the cell long axis, as observed, but also fits very well the measured dependence of the nuclear aspect ratio as function of the cell width.
Ultimately, only one model recapitulates all characteristics of nuclear positioning in VL muscle cells. It suggests that, nuclei repel each other and the cell boundary with forces decreasing with the distance. Our data suggest a simple molecular mechanism, which generates MT pushing forces, either by MT polymerization, or by MT interactions via kinesin motors on the nuclear envelopes and cell cortex.
We support the computational screen of the simple models, in which the nuclei interact as particles by isotropic and deterministic forces, with simulations of a detailed agent-based mechanical model, in which we simulate hundreds of MTs undergoing dynamic instability, bending and pushing on the nuclei and boundary with elastic forces. More importantly, the agent-based simulations generate the single- and double-file nuclear patterns in narrow and wide cells, respectively, as observed and as predicted by the simple models.
Note that each simulation of the microscopic model took hours up to many days on an Linux machine with a Intel Core i processor. As such, parameter exploration of the detailed models, or testing whether they reproduce subtle observed data features, is prohibitive.
In the future, we plan to use more sophisticated mathematical methods [ 44 ] of solving the inverse problems—inferring the models from the data. While the involvement of MTs and molecular motors in the nuclear positioning is firmly established, we do not provide direct proof that a mechanical force balance is the main mechanism of nuclear positioning.
Another possibility is that there is a preexistent, perhaps morphogen-governed, pattern in the cell, and that MTs simply tether the nuclei to special locations in this pattern. Relevant to this thought is the fact that small nuclear clusters aggregate at neuromuscular junction in mammalian cells. However, functioning muscle cells contract, and it is likely that the actomyosin contraction forces are orders of magnitude greater than the MT-based forces.
Thus, it is hard to imagine that MT asters are sufficient to resist nuclear displacement during muscle contractions, and additional nuclear tethers might be involved in maintainaing an established pattern [ 15 ]. Future in vivo experiments, including genetic and biophysical manipulations and live cell imaging, will be required to investigate nuclear positioning in contracting muscle cells.
However, we note that our model generates specific, testable predictions about the nuclear pattern in cases where the cells acquire unusual shapes and sizes or contain variable numbers of myonuclei.
Another intersting aspect of muscle biology that could benefit from our modeling approch is the initial positioning of nuclei in developing embryonic muscle fibers. In the early embryonic muscle cells in Drosophila , after myoblast fusion, the nuclei initially cluster together, then split into two clusters that segregate to the cell poles, and finally spread along the cell length [ 16 ]. It remains to be tested if a force balance model can explain these dynamics. Even more challenging is the problem of coupling of the cell growth, shape change, and protein synthesis with the dynamics of nuclear numbers, positions, sizes and transcriptional activity.
These essentially 3D nuclear patterns require special studies. Active, non-random nuclear positioning has been attracting increasing attention lately [ 45 ]. In a number of recent studies, force generated by MTs and motors were shown to be crucial for nuclear positioning and movement [ 24 , 46 , 47 ]. Zallen, SKI. Whole larvae were mounted in ProLong Gold antifade reagent Invitrogen. Quantification of confocal z -projections was performed using standard ImageJ and Matlab measurement tools.
VL3 and VL4 cells were traced by hand, based on phalloidin labeling. Nuclear centroids were used to calculate nearest neighbor distances. Cell widths and lengths heights were defined as average widths and lengths of the measured boundary.
Nuclear x , y positions were transformed onto positions in a rectangle using a mapping that preserves relative distances from the boundary. Nuclear pattern of both experimental and simulated origins were categorized into single file SF , double file DF or neither using a histogram of the relative nuclear x -positions with 7 equally spaced bins. Parameters are shown in Table 1. K1: All nuclei centroids have to be at least r away from all cell sides and poles.
K2: All nuclei centroids have to be at least 2 r apart. The last criterion avoids counting random patterns as false-positive compare Fig 1E. Candidate models have to lead to valid patterns in both cell geometries, but not for the same parameters this avoids missing good models.
Simulation details are given below. The curves shown in Fig 4A correspond to the parameters that minimize that error. This yielded a score between 0 and 4 for each model and pair c N , c S , the color in the Fig 4B represents this score.
Now the normalized y -positions of all cells were collected using all SF y -positions for the SF auto-correlation analysis, and separating y -positions of nuclei right, and left of the middle of the cell for the DF correlation analysis.
For the final histograms a bin spacing of 0. To determine equilibrium positions, Eq 1 was solved on a rectangular domain using Matlabs ode solver ode15 , a variable-step, variable-order solver. To model finite size effects of nuclei, a size exclusion term was added in Eq 1. For two nuclei whose centroids are a distance d apart, it takes the form. For size exclusion effects between nuclei and the cell boundary, 2 r was replaced by r.
Codes are available upon request. The simulation software Cytosim Ver. The configuration files are available upon request. This file contains three sections: 1. Effect of internuclear friction. This section describes the modeling and simulation of nucleus-nucleus friction. Attraction-repulsion internuclear forces. Comparison of numerical and analytical results for MT-mediated forces. This section compares the shape of a single clamped, confined microtubule, as well as the forces it creates as computed by Cytosim and using an analytical approximation.
The video compares the agent-based, stochastic simulations in Cytosim left with an interacting particle simulation right in the thin, VL4 type cell. Microtubuli are shown as white lines, nuclei as red solid circles, the distance dependent force in the interacting particle model is symbolized by shades of red and yellow.
The video compares the agent-based, stochastic simulations in Cytosim left with an interacting particle simulation right in the wide, VL3 type cell. We thank the J. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. National Center for Biotechnology Information , U. PLoS Comput Biol. Published online Jun Oleg A Igoshin, Editor. Author information Article notes Copyright and License information Disclaimer.
Received Mar 9; Accepted May This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
This article has been cited by other articles in PMC. S1 Video: Agent-based and interacting particle model—Thin cell. S2 Video: Agent-based and interacting particle model—Wide cell. Abstract Many types of large cells have multiple nuclei. Introduction One of the fundamental challenges of cell biology is to define principles of spatial organization of the cell [ 1 ], and, in particular, to unravel the mechanisms that control the position, size, and shape of organelles.
Open in a separate window. Fig 1. Positioning of the myonuclei in Ventral Longitudinal VL muscles 3 and 4. The interacting particle model We make the following assumptions A for the interacting particle models.
Fig 2. Force types and force screen. Forces To complete the system of Eq 1 , we specified the distance dependence of forces f , g S and g P. Force screen structure We found that the screen works best if executed in two filtering steps, which reduced the number for initial models from to 12 Filter 1 and then further to two Filter 2. Fig 3. Sample gallery of the spatial nuclear patterns produced in Filter 1. Table 1 Parameters used for the two filtering steps and the calibration step.
Second filtering step To further reduce the number of models and test for biological relevance, we used the positioning results of a cell screen. Fig 4. Elimination steps in Filter 2. Two model classes result from the screens The two filtering steps resulted in just two model classes that can predict robust nuclear spreading along the cell long axis and correct behavior of x -positions with respect to cell width: Model Class 1: The internuclear forces are repulsive and decrease with internuclear distance.
Calibration To calibrate the two model classes i. Fig 5. Pattern along the x - and y -axis. Transition from single to double file in the nuclear pattern An intuitive explanation for the transition between the SF and DF patterns suggested a way to derive approximate analytical expressions for the x -coordinates of the nuclear positions as functions of the cell width and of the nuclear spacing along the y -direction.
Fig 6. Nuclear file formation and nuclear shapes. Comparison to agent-based, stochastic simulations The screen of the interacting particle models resulted in a single model that fits the data best. Table 2 Parameters in the agent-based, stochastic simulation using Cytosim.
Parameter Values used Comment General time step 0. Calculating distance-dependent forces from agent-based simulations We used the microscopic model, first, to simulate the interactions of a nucleus with a cell boundary in isolation. Fig 7. Microscopic simulation using Cytosim. Agent-based and interacting particle simulations match closely In a multinucleated cell, the assumption of pair-wise additive forces could be violated through a variety of factors.
Image processing and quantification Quantification of confocal z -projections was performed using standard ImageJ and Matlab measurement tools. Filter 1 Parameters are shown in Table 1. Filter 2 Parameters are shown in Table 1. Simulation: Interacting particle model To determine equilibrium positions, Eq 1 was solved on a rectangular domain using Matlabs ode solver ode15 , a variable-step, variable-order solver.
Simulation: Stochastic agent-based model The simulation software Cytosim Ver. Supporting information S1 Text This file contains three sections: 1.
PDF Click here for additional data file. S1 Video Agent-based and interacting particle model—Thin cell. AVI Click here for additional data file. S2 Video Agent-based and interacting particle model—Wide cell. Acknowledgments We thank the J. Data Availability All relevant data are within the paper and its Supporting Information files.
References 1. Building the cell: design principles of cellular architecture. Satellite cells are normally quiescent but become activated and start proliferating in response to growth factors. Since satellite cells are required for skeletal muscle regeneration Robertson et al. At this time point, satellite cells have become activated and are proliferating and beginning to differentiate. We next analyzed the ability of satellite cells to fuse and form new myofibers.
When myofibers form after injury, their nuclei are centrally localized, facilitating identification of regenerating myofibers. The number of centrally nucleated myofibers in the core of the injury 7 d after injury was counted as described in Materials and Methods. At this time point, the area of injury is completely filled with these regenerating myofibers. As seen in Fig. Therefore, satellite cell function appears normal in the early stages of muscle regeneration in the absence of NFATC2.
C A comparison of hematoxylin and eosin—stained sections of regenerating TA muscles at day 25 after injury demonstrates the smaller size of the mutant myofibers top.
The CSA of regenerated myofibers in the central region of the lesion was determined for various time points after injury bottom. To determine if NFATC2 regulates further growth of regenerating myofibers, the CSA of regenerating myofibers was measured at different times after injury. As further growth of the regenerating myofibers occurs, the CSA of myofibers in the mutant at each time point is significantly decreased compared with wild-type Fig.
Thus, the function of NFATC2 is required for growth of myofibers as suggested by our previous in vitro studies Abbott et al. Myofiber growth is dependent on both nonmuscle and muscle cells. A small amount of myogenin is detected in both types of myoblasts, presumably due to spontaneous differentiation in the cultures.
A portion of a Coomassie-stained gel demonstrates relative protein loading. C The number of nuclei within individual myotubes at least two nuclei was counted. Myotubes were grouped into two categories, and the percentage of myotubes in each category was determined. Since myofiber size correlates with myonuclear number Allen et al. To clearly define the nuclei within myotubes, the myotubes were immunostained for EMyHC and the number of nuclei within individual myotubes at least two nuclei was counted.
The percentage of the myotubes in each category was calculated. In addition, an increased number of myotubes occurs in the mutant cultures data not shown. To confirm that the absence of NFATC2 is responsible for the observed defects in myotube size and nuclear number in vitro, two types of experiments were performed. First, the expression levels of other NFAT isoforms in the mutant cells were analyzed to rule out compensatory changes that could contribute to defects in myotube size.
NFATC4 was not detected in either genotype data not shown. These results are consistent with the results of Ranger et al. NFATC4 is not expressed in myoblasts of either genotype data not shown.
Cells were induced to differentiate, and luciferase assays were subsequently performed. Data are reported as fold increase in luciferase activity over control cells. D The number of nuclei within individual myotubes at least two nuclei was counted for each of the retrovirally infected cultures.
Myotubes were grouped into two categories as in the legend to Fig. Myoblasts infected with either control or NFATC2 retroviruses were induced to differentiate, and luciferase activity was determined. No significant difference is observed in nuclear number of wild-type cells infected with either control or NFATC2 retrovirus data not shown , suggesting that overexpressing NFATC2 in wild-type muscle cells does not affect size or nuclear number of myotubes.
Since fusion of myoblasts and thus addition of myonuclei is required for growth of mammalian myofibers Darr and Schultz ; Rosenblatt and Parry ; Mozdziak et al. A A representative wild-type myofiber immunostained with an antibody against dystrophin red and stained with DAPI blue illustrates the myonuclear number assay.
Arrow indicates a myonucleus within the dystrophin border, and arrowheads indicate nuclei outside the myofiber. We have shown previously that individual NFAT proteins translocate to the nucleus of muscle cells at specific stages of myogenesis Abbott et al. To determine if this defect is due to impaired myofiber growth, we examined regenerating skeletal muscle as a model for myofiber growth. Defects in myofiber growth are intrinsic to muscle cells since cultured muscle cells lacking NFATC2 form small myotubes with few nuclei.
Muscle growth results from an ordered sequence of events. Withdrawal of mitogens in vitro causes myoblasts to exit the cell cycle and activate the expression of differentiation specific genes. Subsequently, myoblasts migrate toward one another, elongate, align, and through cell—cell interactions fuse together to form a multinucleated cell.
Multiple proteins are required for migration and fusion of myoblasts such as integrins and other cell adhesion molecules, metalloproteases, and phospholipases Knudsen In addition, increases in intracellular calcium Constantin et al. Once myotubes form, additional myoblasts fuse with the myotube, and the myotube grows in size. Postnatal muscle growth in vivo is also characterized by myoblast fusion with myofibers, leading to an increase in myonuclear number. Numerous studies indicate that myoblasts are critical for muscle growth in vivo.
When the proliferative capacity of myoblasts is attenuated, increases in myonuclear number and myofiber size are blocked in growing rats Darr and Schultz ; Rosenblatt and Parry ; Mozdziak et al.
Additionally, the lack of myoblast growth factors such as leukemia inhibitory factor Kurek et al. Thus, numerous molecules contribute to proper muscle cell size. The decrease in the number of nuclei is correlated with a decrease in myotube size, implicating an NFATC2-dependent pathway in the control of myotube size. Thus, the downstream targets of NFATC2 allow the fusion of differentiated muscle cells with newly formed myotubes and the subsequent growth of the myotube.
Based on our results, we present the model outlined in Fig. Increases in intracellular calcium lead to the activation of calcineurin.
NFATC2 may either directly or indirectly regulate gene transcription of a cell surface protein pathway 1 that mediates cell—cell interaction or cell fusion between mononucleated muscle cells and newly formed myotubes.
Evidence exists for cell surface proteins that mediate the fusion of myoblasts with myotubes. The integrin very late antigen 4 on multinucleated muscle cells and its counterreceptor vascular cell adhesion molecule 1 on myoblasts are thought to mediate myoblast fusion with myofibers during development Rosen et al.
In addition, glycoproteins may mediate myoblast—myotube interactions to allow fusion and muscle growth, since wheat germ agglutinin can block myoblast fusion with myotubes and decrease myotube size Muroya et al. Alternatively, NFATC2 could either directly or indirectly regulate gene transcription of a secreted protein pathway 2 that recruits differentiated mononucleated myoblasts to fuse with adjacent myotubes.
Secreted factors apparently can regulate the fusion of cells with myotubes. Fibroblasts cocultured with young or old myotubes can acquire myogenic characteristics and fuse with myotubes Breton et al. Though not shown, both pathway 1 and 2 could also mediate myotube—myotube fusion. Specific molecules downstream from NFATC2 that contribute to muscle growth, though not examined in this study, are currently being investigated. Only myoblast fusion is represented in the model.
NFATC2 may regulate a cell surface protein pathway 1 that mediates cell—cell interaction or cell fusion between differentiated muscle cells and newly formed myofibers. Alternatively, NFATC2 could regulate gene transcription of a secreted protein pathway 2 that recruits differentiated muscle cells to fuse with adjacent myofibers. Myotube—myotube fusion has been suggested to occur during regeneration Robertson et al.
Several lines of evidence implicate multinucleated cells as controlling the site and extent of myoblast fusion. During the development of mammalian skeletal muscle, primary myofibers form initially and are followed by the formation of secondary myofibers. Primary myofibers control the site of secondary myofiber assembly, since secondary myofibers form only at the site of innervation on the primary myofiber, independently of the nerve Duxson et al. In addition, the primary myofiber seems to restrict the fusion of secondary myoblasts, whereas the secondary myofiber seems to recruit fusion, since secondary myoblasts fuse primarily with the forming secondary myofiber Harris et al.
This specificity of fusion shares analogy with Drosophila muscle development in which founder myoblasts express dumbfounded, an attractant for myoblast fusion, and recruit fusion-competent myoblasts to fuse with founder myoblasts and not with other myoblasts Ruiz-Gomez et al. Further suggesting that myofibers can control the location of myoblast fusion, myofibers elongate by fusion of myoblasts at their ends during mammalian postnatal growth Williams and Goldspink ; Zhang and McLennan The mechanisms by which myofibers regulate the fusion of myoblasts are unknown but may involve expression of proteins such as those mentioned above.
This control of fusion by multinucleated muscle cells is likely one mechanism by which the size of muscle cells is regulated. Calcineurin has been shown to be involved in skeletal muscle hypertrophy Musaro et al. The inability of NFATC2 overexpression to induce an increase in myotube size and nuclear number in wild-type myotubes suggests that NFATC2 is not involved in skeletal muscle hypertrophy.
Several possibilities exist to explain this lack of effect on growth of wild-type myotubes. This hypothesis is supported by the fact that NFATC2 cannot be activated by a calcium ionophore in mature myotubes Abbott et al. These pathways could involve the activation of additional transcription factors necessary for forming a transcriptional complex with NFATC2.
It forms the contractile component of the digestive, urinary, and reproductive systems as well as the airways and blood vessels. Each cell is spindle shaped with a single nucleus and no visible striations Figure 4. Watch this video to learn more about muscle tissue. In looking through a microscope how could you distinguish skeletal muscle tissue from smooth muscle? The three types of muscle cells are skeletal, cardiac, and smooth.
Their morphologies match their specific functions in the body. Skeletal muscle is voluntary and responds to conscious stimuli. The cells are striated and multinucleated appearing as long, unbranched cylinders. Cardiac muscle is involuntary and found only in the heart.
Each cell is striated with a single nucleus and they attach to one another to form long fibers. Cells are attached to one another at intercalated disks.
The cells are interconnected physically and electrochemically to act as a syncytium. Cardiac muscle cells contract autonomously and involuntarily. Smooth muscle is involuntary.
Each cell is a spindle-shaped fiber and contains a single nucleus. No striations are evident because the actin and myosin filaments do not align in the cytoplasm.
You are watching cells in a dish spontaneously contract. They are all contracting at different rates, some fast, some slow. After a while, several cells link up and they begin contracting in synchrony.
Discuss what is going on and what type of cells you are looking at. The cells in the dish are cardiomyocytes, cardiac muscle cells.
0コメント