Morphological classification based on structure. There are two types of muscle based on the morphological classification system. There are two types of muscle based on a functional classification system.
Types of muscle: there are generally considered to be three types of muscle in the human body. Skeletal muscle: which is striated and voluntary. Cardiac muscle: which is striated and involuntary. Smooth muscle: which is non striated and involuntary.
Skeletal muscle cells are elongated or tubular. They have multiple nuclei and these nuclei are located on the periphery of the cell. Skeletal muscle is striated. That is, it has an alternating pattern of light and darks bands that will be described later.
Cardiac muscle cells are not as long as skeletal muscles cells and often are branched cells. Cardiac muscle cells may be mononucleated or binucleated. In either case the nuclei are located centrally in the cell. Cardiac muscle is also striated. In addition cardiac muscle contains intercalated discs. Smooth muscle cell are described as spindle shaped.
That is they are wide in the middle and narrow to almost a point at both ends. Smooth muscle cells have a single centrally located nucleus. Smooth muscle cells do not have visible striations although they do contain the same contractile proteins as skeletal and cardiac muscle, these proteins are just laid out in a different pattern.
Parallel or fusiform: as their name implies their fibers run parallel to each other. These muscles contract over a great distance and usually have good endurance but are not very strong.
Examples: Sartorius muscle and rectus abdominus muscle. Convergent: the muscle fibers converge on the insertion to maximize the force of muscle contraction. Examples: Deltoideus muscle and Pectoralis Major muscle. These types of muscles are strong but they tie or quickly. There are three types of pennate muscle. Nevertheless, the myonuclei remain appropriately positioned along the cell, although the mechanisms that are responsible for this are not clear.
While the actomyosin network may be involved in nuclear positioning [ 15 ], microtubules MTs , MAPs MT Associated Proteins , and MT-based motors, such as kinesin and dynein, have been shown to play a major role [ 12 , 16 — 18 ]. As examples, in embryos in which MTs are severed in the muscle cell, the central cluster does not split; in many motor mutants, nuclear spreading in the muscle cell is perturbed [ 19 ].
However, the precise mechanisms controlling myonuclear positioning remain poorly understood. Modeling has proven to be very useful in complementing cell biological methods in problems of positioning with, for example, the mitotic spindle [ 20 , 21 ].
Mathematical modeling focused on multinucleated cells and nuclear positioning is in its infancy. Simple conceptual models of nuclei repelling each other were used in [ 22 ] and [ 23 ] to show that such models can explain regular distribution of nuclei in muscle cells and in the Drosophila blastoderm syncytium.
Detailed mechanical simulations were done in [ 24 ] to understand multiple nuclear movements in multinucleate fungus Ashbya gossypii. Here, we use computational modeling to understand the mechanisms regulating nuclear positioning in Drosophila larva muscle cells.
We hypothesize that nuclear positioning is a result of a MT-motor based force balance. Rather than assuming the nature of this force balance, we screened multiple computer-generated forces by comparing the spatial nuclear patterns that they predict to quantitative microscopy data from biological specimens.
We then simulated a detailed agent-based model to confirm the predictions of the screen. One model explains all biological data, including many subtle patterns of multi-nuclear positioning. Based on this model we propose that myonuclei are positioned by establishing a force balance via MT-mediated repulsion. Representative confocal microscopy images of fixed samples with fluorescently labeled muscle cells actin , myonuclei lamin, Hoechst and microtubules alpha-tubulin are shown in Fig 1A—1C.
VL muscles are flat rectangular cells with nuclei located on one cell side. We employed the following terminology: cell length is the dimension along the long axis of the cell y -direction , cell width is the dimension along the short axis of the cellular rectangle x -direction Fig 1B.
We referred to the edge of the z -projection of the cell as the cell boundary, and distinguished between cell sides long segments of the rectangular shape and cell poles short segments of the rectangle. We defined the subcellular localization of nuclei and measured several geometric parameters relevant to nuclear positioning, including nuclear numbers, shapes, nearest neighbor distances, and distances to the cell boundary, and used statistical tools to analyze the data.
A: Left: Drosophila 3 rd instar larva anterior up , right: Dissected larva revealing the somatic musculature, labeled with phalloidin red to reveal actin structures sarcomeres in muscles. D: Frequency of wide vs long nuclei defined using the angle of the major axis of a fitted ellipse. F: Histograms of the nuclear positions along the x - and y -axes for both cell types. The half-length and half-width of the cells were normalized to 1.
G: Average nuclear x -positions relative to the center for each cell as functions of the cell width. The following features of VL muscle cells and nuclei informed our modeling: 1 Both cell types share a similar length VL3: We simulated random nuclear positioning by generating random and independently uniformly distributed x - and y -coordinates of the nuclear centers in rectangular domains.
Analyzing the experimentally measured nearest-neighbor distances between the nuclei showed that the arrangement is not random Fig 1E ; rather, there is a characteristic distance between the neighboring nuclei. In fact, the average nuclear x -position within each cell is a function of the cell width Fig 1G : the nuclear x -position increases with increasing cell width, irrespective of the cell type.
Thus, nuclei in a wide VL4 cell have the same position as those of a VL3 cell of equal width. When MT organization is disrupted, nuclei have been shown to be mispositioned [ 25 ], suggesting that MTs exert forces on the nuclei, and that the resulting force balance is the key to the nuclear positioning.
After assuming or calculating a mean MT-motor force between material objects in the cell as a function of distance between the objects, one can solve equations of motion for the interacting objects via position-dependent force laws.
This leads to solving a system of ordinary differential equations ODEs , the number of which is equal to the number of the objects [ 22 , 23 , 26 , 27 ]. Such models can be simulated rapidly such that many different internuclear force types can be screened.
This leads to solving a large system of partial differential equations PDEs , or a gigantic ODE system, which is much more difficult and computationally expensive than the first approach. While this approach provides more detailed predictions, many motors are involved in the force balance [ 19 ], and it is not clear which combination of the motors generates the force in this context. Further, mechanical characteristics of most of these motors are unknown.
In principle, one could use all possible motor combinations [ 28 ], but the long computation times and high dimensionality of the model parameter space makes parameter scans of the detailed stochastic model impossible [ 29 ]. In this work, we systematically combine the two approaches to suggest mechanisms of nuclear positioning.
We begin by screening various forces using interacting particle modeling and determine which model can be responsible for not only qualitative features of the observed spatial patterns, but also explain quantitatively all the subtle geometry of the nuclear positioning. Such an unbiased and systematic computational screen was needed for a few reasons. First, there are multiple simple and intuitive models that predict roughly uniform nuclear distribution, and choosing between them by a traditional thinking process is vulnerable to psychological biases.
Second, spatial patterns generated by simple forces may be highly complex, counterintuitive, and non-robust [ 30 ]. After this screen, we used learned force characteristics to inform a detailed agent-based stochastic model with explicit simulations of individual MTs to confirm the lessons from the screening.
We make the following assumptions A for the interacting particle models. A1: The cell can be described as a 2D rectangle. A2: The velocity of a nucleus is proportional to the total force acting on this nucleus. This is typical in a viscous friction-dominated environment in cells characterized by low Reynolds numbers [ 20 , 21 , 23 , 24 ]. A3: Nuclei interact with each other via pairwise interactions; further, independent interactions exist between each nucleus and the cell boundary.
A4: All interactions are deterministic, isotropic and result in distance-dependent forces discussed below. A5: The forces are additive: movement of the i -th nucleus at each time moment is determined by the sum of the forces acting on it from all other nuclei as well as from the cell sides and cell poles. Assumptions 1 is well justified by the biological data. With regards to A2, we note that MT entanglement could cause an effective additional internuclear friction [ 23 ], in which the speed of a nucleus is influenced by the speeds of neighboring nuclei.
We tested such a scenario see S1 Text and found that while myonuclear movement is affected by this additional friction, there seems to be very little effect on the final nuclear positions. Since this study is concerned with equilibrium positions only, we decided to omit the internuclear friction; however, it might be an important consideration for future work dealing with transient dynamics.
We discuss the validity of A in Sec Comparison to agent-based, stochastic simulations; here we briefly comment on origins of these three assumptions. Examples of molecular mechanisms shown in Fig 2A invoke forces generated by stochastic processes of MT growth and motor action. Assuming radially symmetric MT and motor distributions on nuclear envelopes, the forces are approximately isotropic. The pairwise and additive character of the forces is based on the scenario in which one set of MTs from one nucleus interacts with the second nucleus, while another MT set from the first nucleus interacts with the third nucleus or the cell boundary; in this case, the force on the first nucleus is equal to the sum of two independent forces from the second and third nuclei respectively.
However, flexible interacting MTs and possibility of the second nucleus being between the first and third one complicate this picture. Only a full stochastic model can address the full complexity. A: Possible molecular mechanisms of nuclear interaction via MT generated forces ; see key bottom left, gray circles represent nuclei. Different forces f positive for repulsive, negative for attractive as a function of distance d are depicted in the lower row.
Greater distances between the nuclei lead to a larger MT overlap assuming long MTs , hence the force is increasing with distance. Depending upon whether the number of MTs or the number of kinesins are limiting, the resulting repulsive force can be decreasing with distance or be distance independent. B: Structure of the force screen: Two filtering steps are followed by a calibration step. Filter 1: The models are filtered by their ability to produce an evenly spread single file SF of nuclei in the thin cell, and double file DF in the wide cell.
We found that the vast majority of the models fail this step and can be discarded; only 12 potential model classes remained. Filter 2: We applied each of the 12 model classes to 14 imaged cells of representative width, height and number of nuclei and examined whether the models can account for two characteristics of the biological data: 1 average nuclear x -position increases with cell width, 2 the model behavior is robust with respect to parameter changes.
After this second filter, only two model classes remained. We denote by f the scalar forces between pairs of nuclei, and by g S and g P the scalar forces between a nucleus and the cell sides or a nucleus and the cell poles, respectively. Since we are only interested in equilibrium nuclear positions in this study, we normalized the effective drag coefficient characterizing the nucleus to unity. Thus, the velocity of the i -th nucleus is equal to the sum of all forces acting on it:.
Here w i k is the normal projection of the 2D position of the i -th nucleus on the corresponding cell boundary, the letters U , D , L , R denote the cell boundary up, down, left and right of the nucleus respectively.
The addition of a size exclusion term is described in the Methods. To complete the system of Eq 1 , we specified the distance dependence of forces f , g S and g P. Fig 2A depicts possible molecular mechanisms motivating different force distance dependencies; the figure legend explains the molecular details. These examples produced either purely repulsive or attractive forces, which can be either increasing or decreasing with distance or are distance-independent.
Additionally, it is reasonable to assume that in some cases when MTs grow only to a certain length , forces may have only a finite reach. To represent the forces mathematically, we used the expressions:.
Of course, the effective forces of interactions mediated by complex MT behavior and multiple motor types may be more complicated: these may be repulsive at some distances and attractive at others, have distance dependencies other than the power laws [ 26 , 28 ], and even have non-monotonic distance dependence [ 31 ]. To address these possibilities, we did numerical experiments with force expressions, including exponents other than those used in the screen, and with functions other than power laws, including exponential functions and others.
We found that all qualitative results of the screen remained valid after such variations. We also experimented with forces that are repulsive at some distances and attractive at others see S1 Text , and found that as long as the long-ranged forces are relatively small, they lead to results similar to those predicted by purely repulsive or attractive forces.
As for more complex possibilities of non-monotonic distance dependence, these are beyond the capabilities of our approach. In the future, if data suggest such possibilities, it will have to be considered then. In other words, each combination of three repulsive or attractive forces, each characterized by being either decreasing or increasing with distance or distance independent, is a model.
In addition, each model is characterized by two relative force amplitude parameters M S , P and by three force range parameters c N , S , P. We also found that the exact form of the pole force has little effect on the nuclear distribution, as far as the pole force has a range much smaller than the cell length.
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. This is followed by a calibration step. The steps of the force screen are summarized in Fig 2B and detailed below.
This allowed us to select and characterize the ultimate model. Tab 1 lists all parameters used. Each model was tested with typical VL3 wide and VL4 thin cell geometries, starting with both randomly placed and equally spaced nuclei. Each simulation was run until an equilibrium was reached and the resulting spatial patterns were evaluated using the following criteria Filter 1 in Fig 2B , legend in Fig 3 , for details see Methods :.
A: Equilibrium positions for the 36 model classes examples. Combinations of forces att: attractive, rep: repulsive, dec: decreasing, inc: increasing, con: constant. We show results of varying the character of internuclear and nuclear-cell side forces. For each force combination, we show predicted patterns in a narrow VL4 and wide VL3 cell black rectangles. Nuclei at their equilibrium positions are shown as blue and red discs for valid and non-valid patterns respectively. Outside of the green box, the models have not passed the first filtering step for any parameter values.
Green box: model classes that passed the first filtering step for some parameter combination positions shown represent valid patterns. For one particular combination grey, dashed box , further details of scanning the parameter space are shown in B.
Blue nuclei indicate valid patterns. For parameters see Table 1 , for details see Methods. Fig 3 shows the result of the first filtering step. We found that 24 of the 36 modeling classes did not satisfy all five criteria imposed by the biological data for any values of the model parameters Fig 3A. Twelve model classes passed Filter 1 successfully, in the sense that we found parameter values for these model classes for which all five criteria imposed by the data were satisfied examples shown in in Fig 3B.
These 12 classes, C, were characterized by the internuclear and nucleus-cell side forces; pole forces played a lesser role. For the nuclear-cell side forces, a variety of forces are possible: attractive forces increasing with distance spring-like forces that reach across the width of the cell, as well as all types of repulsive forces. For the distance independent repulsion or repulsion increasing with distance, the side forces need to have a very specific reach of about half the width of a typical VL4 cell.
If the reach is above or below that, nuclei in VL4 cells do not align along the center of the cell. To further reduce the number of models and test for biological relevance, we used the positioning results of a cell screen.
Specifically, we used the dimensions and nuclear numbers measured in 14 representative cells see the list and parameters in Table 1 , applied the 12 remaining model classes to each of these 14 cells, and evaluated the nuclear positions according to two criteria: 1 Does the average absolute nuclear x -position increase with cell width? Fig 4A 2 How robust is the model with respect to changes in parameter values Fig 4E. Focusing on the interplay between internuclear and side forces, we fixed the pole forces to short-ranged repulsive forces compare Table 1.
To apply the first criteria, we used the sets of parameters force magnitude and range for each of the modeling classes that provided a minimal error in predicting the computed average absolute nuclear x -position of the experimental data Fig 4B. The results shown in Fig 4A led to the exclusion of model classes C, since these models predict that the average absolute nuclear x -position initially decreases with cell width, in contrast to the biological data.
An intuitive explanation for this behavior is depicted in Fig 4C for the example of repulsive, increasing with distance side forces, with a finite range: For very thin cells the side forces from the right and left overlap, creating a region in the middle of the cell, where the nucleus feels an effective decentering force. Only a very specific cell width will promote centering explaining the dip in the width vs x -position plot. Finally, for wide cells, nuclei near the cell center will feel no side forces, however, since they will be pushed by other nuclei, this will again lead to an x -position away from the center.
In comparison, repulsive side forces, decreasing with distance Fig 4D will always promote centering, however as cells get wider, these forces reduce and eventually neighboring nuclei succeed in pushing the nucleus out of the center. 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. Skeletal muscle is composed of very hard working cells.
Which organelles do you expect to find in abundance in skeletal muscle cell? 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.
Rather than assuming the nature and identity of the forces, we simulated various types of forces between the pairs of nuclei and between the nuclei and cell boundary to position the myonuclei according to the laws of mechanics. We started with a large number of potential interacting particle models and computationally screened these models for their ability to fit biological data on nuclear positions in hundreds of Drosophila larval muscle cells.
This reverse engineering approach resulted in a small number of feasible models, the one with the best fit suggests that the nuclei repel each other and the cell boundary with forces that decrease with distance.
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