The organization of muscle is the product of functional adaptation over several length scales spanning from the sarcomere to the muscle bundle. One possible strategy for solving this multiscale coupling problem is to physically constrain the muscle cells in microenvironments that potentiate the organization of their intracellular space. We hypothesized that boundary conditions in the extracellular space potentiate the organization of cytoskeletal scaffolds for directed sarcomeregenesis. We developed a quantitative model of how the cytoskeleton of neonatal rat ventricular myocytes organizes with respect to geometric cues in the extracellular matrix. Numerical results and in vitro assays to control myocyte shape indicated that distinct cytoskeletal architectures arise from two temporally-ordered, organizational processes: the interaction between actin fibers, premyofibrils and focal adhesions, as well as cooperative alignment and parallel bundling of nascent myofibrils. Our results suggest that a hierarchy of mechanisms regulate the self-organization of the contractile cytoskeleton and that a positive feedback loop is responsible for initiating the break in symmetry, potentiated by extracellular boundary conditions, is required to polarize the contractile cytoskeleton.

How muscle is organized impacts its function. However, understanding how muscle organizes is challenging, as the process occurs over several length scales. We approach this multiscale coupling problem by constraining the overall shapes of muscle cells to indirectly control the organization of their intracellular space. We hypothesized the cellular boundary conditions direct the organization of cytoskeletal scaffolds. We developed a model of how the cytoskeleton of cardiomyocytes organizes with respect to boundary cues. Our computational and experimental results to control myocyte shape indicated that distinct muscle architectures arise from two main organizational mechanisms: the interaction between actin fibers, premyofibrils and focal adhesions, as well as cooperative alignment and parallel bundling of more mature myofibrils. We show that a hierarchy of processes regulate the self-organization of cardiomyocytes. Our results suggest that a symmetry break, due to the boundary conditions imposed on the cell, is responsible for polarization of the contractile cytoskeletal organization.

Funding: This work has been supported by the Nanoscale Science and Engineering Center of the National Science Foundation under NSF award number PHY-0117795, the Harvard Materials Research Science and Engineering Center under NSF award number DMR-0213805, the DARPA Biomolecular Motors program, and NIH grant 1 R01 HL079126 (KKP). Mark-Anthony Bray acknowledges salary support from a UNCF-Merck Science Initiative postdoctoral fellowship. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Myofibrils mature in a force-dependent manner [7] , [8] , [9] , suggesting that the contractility of a cell may play an important role in polarizing the myofibrillar network. This has been shown in nonmuscle cells where the cytoskeletal architecture within a geometrically-defined microcompartment becomes polarized with increasing tractional forces [10] , [11] . Thus, we hypothesized that geometric cues in the extracellular matrix (ECM) can organize the intracellular architecture and potentiate directed myofibrillogenesis. Because of the difficulty in identifying de novo sarcomeres in primary harvest muscle cells in culture, one strategy for studying myofibrillogenesis is to coax the disassembly and reassembly of myofibrils by forcing myocytes to assume shapes that are not commonly observed in vivo using engineered substrates in vitro [10] , [11] . To guide these experiments, we developed a computational model of myofibrillar patterning to show the sensitivity of the intracellular architecture to the extracellular space. With these tools, we sought to understand the critical events in the global assembly and organization of the contractile apparatus in cardiac myocytes. By comparing experimental results with our computational model, we were able to elucidate the role of maturing myofibrils, their parallel coupling, and their functional attachment to the focal adhesion assembly and how these processes are guided spatially by the boundary conditions imposed on the cell. After determining the roles of these parameters in myofibrillogenesis, we then expanded our model to test the functional implications of these architectures. We developed a novel method for micropatterning on soft substrates and were able to engineer myocyte shape on substrates that would allow us to measure the contractility of these artificial shapes and compare them with the model results. Together, these results suggest that the self-assembly and -organization of the contractile apparatus is facilitated by a symmetry-breaking event that is potentiated by either a geometric cue in the extracellular space or a random event in the intracellular space.

During biological development, evolving forms are marked by distinct functionalities. An interesting example is the organization of myofibrils in striated muscle cells. As the myocyte matures, the myofibrils are rearranged from an irregularly dispersed pattern into tightly organized bundles spanning the length, rather than the width, of the cell [1] . Although assembly of the myofibril from its molecular constituents has been extensively investigated [2] , [3] , [4] , how myofibrils build this specialized architecture and its functional consequences remains unanswered. This is important because changes in muscle structure accompany not only morphogenesis, but also pathogenesis [5] , [6] .

Results

Qualitative Description of the Model Our theoretical approach focuses on the interaction between the myofibril and the ECM, as well as adjacent myofibrils (Fig. 1). Inherent to our model are two key assumptions: 1) the force that the myofibrillar bundle exerts on the substrate is fiber length-dependent [12] and 2) adjacent myofibrils affect each other to facilitate lateral coupling, which is akin to them exerting torque on each other. We have modeled only the maturation of cytoskeletal structural elements responsible for contraction and integrin binding to the ECM. We define these components using coarse-grained variables that are experimentally observable. This eliminates the computational complexity required to model detailed molecular interactions and the effect of different protein isoforms. The nomenclature for the immature and mature versions of the myofibril vary with different qualitative models (reviewed by Sanger and colleagues [2]). Here we refer to the immature state as the premyofibril, and the quasi-mature state as the nascent myofibril [1], [13]. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 1. Schematic representation of myofibril reorganization in a 2D myocyte. (A) red: actin; blue: nucleus; green: FAs. The FAs can spread throughout the ECM island (outlined by solid black island). (B) Net force (F) exerted on bound integrins, as determined by the sum of all forces exerted by the anchoring premyofibril vectors, recruits free integrins and promotes growth of FAs. For the purposes of modeling the bound integrins connected to premyofibrils are labeled ρ p (r). (C) Continued recruitment of free integrins to the growing FA at the cellular corners is associated with enhanced bundling of the premyofibrils and subsequently increased traction. (D) Built upon the premyofibrillar network, the nascent myofibrils align in parallel and develop into a fully organized bundle, further amplifying local force to result in FA maturation. For the purposes of modeling the bound integrins connected to nascent myofibrils are labeled as ρ n (r). (E) Bound integrins with zero net force cannot recruit free integrin and are disassociated from the membrane, leading to release of the attached fiber (F). Consequently, contractile fibers on shorter axes (G) are less bundled than that following the longest diagonal of the cell. (H) Qualitative schematic of model implementation algorithm. https://doi.org/10.1371/journal.pcbi.1001088.g001 Our mathematical approach differs from others [14], [15] in that we incorporate focal adhesion (FA) kinetics, mutual alignment of adjacent contractile fibers, and the dependence of contractile forces on fiber length [16]. The variables used in our approach are: (1) the density of bound and unbound integrin, and , respectively; with the bound integrins connected to premyofibrils and nascent myofibrils labeled as and , respectively; (2) the net force exerted on the bound integrin, ; (3) the local density, , orientation, , and the orientational order parameter, , of the premyofibril network and the nascent myofibril network; and (4) the resultant 2D stress field exerted by the cell on the substrate, T. Previously, we reported [17] that when cardiac myocytes are constrained on 2D islands, their vertical dimension, orthogonal to the plane of the culture surface, is uncontrolled. In that study, we reported that myofibrils are predominantly located under the nucleus, in a plane parallel to the culture surface. However, as that study also showed, several layers of myofibrils may be present, and the nucleus and microtubule organizing center may represent an obstacle to a symmetrical array of myofibrils in the thicker regions of the cell. Our model and analysis is restricted to the 2D intracellular plane closest to the culture surface. Instead of solving the steady state for all of the variables, we numerically simulated their spatiotemporal profiles. This allows us to trace the effect of local symmetry-breaking events such as the mutual alignment of fibers on myofibrillar patterning, which cannot be easily predicted by conventional steady-state analysis. The local symmetry-breaking event may result from a static cue or a transient perturbation. In our simulation, we began with randomly distributed densities of the unbound integrin, unless fitting parameters, in which case we examined several sets of initial conditions. The unbound integrin can initially become bound through a random process, with the rate proportional to its local concentration. The fraction of bound integrins connected to the fibrils is modeled as an adsorption process, and is calculated using the Langmuir isotherm. The force exerted between FAs is assumed to be proportional to the product of fiber connections at each site [16]. The net force at a local FA is computed by integrating the tension contributed by all connected contractile elements (Fig. 1A). The net force governs the growth rate of local FAs, which in turn modulates the premyofibril network [18], [19]. The assembly of FAs and the bundling of its associated fibers is coupled by a positive feedback loop via forces exerted on the FA [16], [18]. As a consequence of the positive feedback, when the net force on a FA is not zero, both the FA and its associated fibers are structurally reinforced (Fig. 1B–D) [20]. If the net force is zero, the bound integrins will disassemble at each time step and disassociate the attached fibers (Fig. 1E–G) [18], [21]. As time lapses, the premyofibrils are converted to the nascent myofibrils. The local orientation of the nascent myofibril is primarily determined by the antecedent premyofibril network, but also can be modulated by adjacent myofibrils due to their lateral coupling [1], [22]. In some cell shapes, polarization of the myofibrillar array can only be achieved by the lateral alignment of adjacent myofibrils, which occurs at a much slower time scale than that of fiber assembly [1], [22]. The effect of the lateral coupling is modeled as a biasing potential field that distributes the free integrins, such that the nascent myofibrils are moved towards each other through the course of normal integrin recycling. To visualize the amount of parallel, or lateral, coupling of the fibers, we define a variable, ψ, which varies from zero for no local coupling, to unity for the maximal local coupling. The model's calculations are ordered as depicted in Fig. 1H.

Model versus Experiment: The Architecture of a Stair-Shaped Myocyte To fit the parameters of the computational model, we chose an uncommon cell shape, a stair-shaped myocyte, that we could model computationally in silico and repeatably in vitro with cell engineering techniques (Fig. S1). The parameters were fit on a variety of initial conditions (Fig. S2) such that the steady state results were the same for each. In Fig. 2A we show the temporal results for an initial condition with a random distribution of free integrins. Initially, there are no fibers in the cell, as no integrins are bound (Fig. 2A ). The geometrical symmetry of the stair-shape cell potentiates the initial appearance of fibers predominantly along the diagonal. As the fibers form, the fiber density is mostly uniform throughout the cell, as evident from the line segment thickness (Fig. 2A ). When the nascent myofibrils form and begin to laterally couple, they are distributed diffusely within the cell (Fig. 2A ). As time progresses, the positive feedback increases, i.e. greater number of fibers produces a greater force which drives the clustering of bound integrins and fibers. As a result, the myofibrils achieve a distribution very similar to the steady state (Fig. 2A ). For the rest of the simulation the nascent myofibrils mutually align and exhibit greater degrees of parallel coupling (Fig. 2A ). Myocytes were cultured on stair-step shaped islands for three days and then stained against actin filaments (Fig. 2B). At equilibrium, most nascent myofibrils are coupled and aligned with the major diagonal, as shown experimentally in Fig. 2B and in simulation (Fig. 2A ). The parallel coupling of the nascent myofibrils emerges later in the simulation, as suggested by previous reports [1], [21], [22], [23]. In summary, the simulated dynamics visualized for nascent myofibril bundling and realignment show that well-aligned myofibrils first occurred in the center of the cell, followed the longest diagonal, and recruited additional adjacent fibers to form a bundled, parallel arrangement. PPT PowerPoint slide

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larger image TIFF original image Download: Figure 2. Simulated dynamics of myofibril organization and immunostaining of actin alignment. (A) Simulated results for the dynamic profile of myofibril organization in a stair-step-shaped myocyte. Red lines represent the myofibrils, with thicker lines representing regions of denser myofibrils. The grey color scale represents the amount of local parallel coupling of the nascent myofibrils; color values are in arbitrary units normalized to the highest values. As we start with a random distribution of free integrins, initially there were no fibers. The geometrical symmetry break in the stair-cell is so strong that for random initial conditions the fibers generally align with the major diagonal as soon as they are formed. However, nascent myofibrils become latterly coupled throughout the cell as evident by the diffuse grey shading at . As time elapsed, the nascent myofibrils reorganized and oriented themselves along the longest cellular diagonal, and coupled to each other greatly increasing parallel coupling. The steady state fiber organization matches the experimental results. (B) Immunostaining of the actin network from a myocyte with similar shape agrees with the numerical prediction; scale bar: 10 µm. https://doi.org/10.1371/journal.pcbi.1001088.g002