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Multisarcomere 3D Explicit Stochastic Model

MUSICO (MUscle SImulation COde) provides a comprehensive multiscale computational framework designed to simulate a wide variety of experimental muscle behaviors, for multipurpose use by the muscle community. The most important uses are to interpret experimental data, identify gaps in our current understanding, and provide information that can be used in the development of novel hypotheses and the design of critical new experiments to test them. The modular structure of MUSICO allows quantitative assessment of the effect of variations in sarcomere geometry, incorporation of new models with kinetic and structural details of the actomyosin cycle, provide more detailed descriptions of thin filament regulation and provide new knowledge on the role of auxiliary muscle proteins in muscle functions. The platform explicitly incorporates the 3D sarcomere structure with extensible actin and myosin filaments, along with various actomyosin cycles, thin filament regulation models and protocols of biochemical and mechanical loading. The modular structure of the MUSICO platform permits various combinations of modules.

Multiscale modeling integrates the action of molecular (crossbridge) forces into a larger scale 3D sarcomere structure. A unified description of muscle function has been achieved by using a finite element model in which all geometrical factors and the constitutive elastic properties of filaments and crossbridges are compounded in the system’s stiffness matrix. On the molecular scale, the kinetics of the underlying biochemical reactions defines the state rate transition matrix that integrates the biochemical reactions of the actomyosin cycle and its           regulation, and the regulation of cell activity by undefined messengers or excitation-contraction-coupling pathways. Integrating these two aspects of the model, which operate on different length scales, and the activation and boundary conditions (         transients and external forces or displacements) requires iterative numerical procedures to predict the effects of strain-dependent modulation of the key biochemical reactions over multiple scales and nonlinear finite element analysis.

Sliding filament models of muscle contraction interconnect the actomyosin ATPase cycle, thin filament regulation and the structural arrangement of myosin and actin filaments into the sarcomere lattice. These models of various levels of complexity predict the dynamic response of muscle to        activation and external boundary loads. The modular structure of the MUSICO platform is designed to address many different scientific problems by adding new modules or adapting existing modules for the specifics for each problem. The flexible structure of the platform permits use of various combinations of actomyosin cycles, thin filament regulatory models, sarcomere structures and protocols of biochemical and mechanical loading

The spatially explicit stochastic model of muscle contraction (MUSICO), that takes into account the interaction of each myosin head with a few nearest binding sites on actin in the three dimensional sarcomere lattice. In this lattice the relative distance between a myosin head and the adjacent binding site on actin is defined by four factors: (i) the axial displacements along myosin and actin filaments, (ii) the transverse spacing between myosin and actin filaments, (iii) the angle     defining relative position of myosin to actin filament, and (iv) the angle     defining how much a myosin head needs to turn in order to reach an actin monomer in the correct orientation. The spatial strain dependence is currently implemented for four actomyosin cycles: two state Huxley 1957, three state Duke 1999 & Daniel et al. 1998 models, four state Smith 1998 and the most comprehensive nine state Smith et al., 2008 model. Modular structure of the platform allows implementation of other relevant acto-myosin cycles without significant changes in other parts of the platform.

Models of thin filament regulation. Contraction in striated muscle is regulated by calcium and the actin filament-associated proteins tropomyosin (Tm) and troponin (Tn). In relaxed muscle, the absence of        induces Tn to hold Tm in an azimuthal position that sterically blocks myosin binding sites on F-actin and, therefore, inhibiting muscle contraction and force generation. In contrast, during muscle activation, the presence of         changes the position of the TmTn complex, moving it azimuthally to a position where it temporarily inhibits strong myosin binding to actin, thus favoring actin-myosin interaction and initiating muscle contraction. When a myosin head strongly binds to an actin filament, it further displaces the TmTn complex, generating a larger angular displacement, which facilitates binding of nearby myosin heads; this is a potential origin of the cooperativity in myosin binding observed during muscle contraction.

The calcium dependence of myosin binding to the fully-regulated actin filament can be predicted if the actin affinity of TnI is a known function of the free calcium concentration. The binding of TnI to actin is down-regulated by the binding of two molecules of         to TnC in skeletal muscle or one molecule of           in cardiac muscle.  We have implemented in MUSICO two stage model: one for calcium binding to TnC and subsequent loss of affinity to actin and the other to account for the regulation of myosin binding by tropomyosin current position. So far, modules for        binding to TnC and kinetics of TnI interaction with actin for skeletal and cardiac muscles are implemented in MUSICO .

Calcium binding to Tn and Tn interaction with actin in skeletal and cardiac muscle. In skeletal muscle one or two calcium ions bind to TnC changing affinity of TnI to actin (denoted as red vertical arrows) and attachment detachment of TnI to actin (denoted as green horizontal arrows). In cardiac muscle only one calcium ion binds to TnC and the scheme reduces to only four states.

Vertical arrows denote kinetics of         binding to actin and horizontal TnI interaction with actin where left columns (red) denote states where TnI bound to actin and right columns denote states where TnI is free where associated tropomyosin can move (azimuthally) on actin surface and permit myosin binding to actin. The implemented schemes allow dynamic changes in calcium concentrations and the calcium transients can be prescribed, for example during physiological twitch contraction.        

For the second stage of thin filament regulation, cooperative McKollp-Geeves three state model and continuous flexible chain (CFC) model are implemented in MUSICO. The details of these models are described here.

Lattice structural arrangements. Modular structure of MUSICO platform permits different structural arrangements in multi sarcomere structures. So far we implemented hexagonal lattice with myosin filaments with three myosin molecules per crown and with four myosins per crown in flight muscle. Axial arranges can be prescribed for a specific geometry of muscle including periodicity of crowns, actin monomer spacing and arrangement of regulatory proteins. The distance between the actin and myosin filaments can vary with the sarcomere length and from muscle type to muscle type, and experimental conditions, for example intact vs. skinned muscles.

Several models are currently implemented in the platform including structural ancillary proteins nebulin, titin and myosin binding protein C (MyBP-C). Due to the complex nature of role of these proteins in muscle function and diverse experimental data, this part of platform is still under development permitting implementation of experimental findings or variations as they appear.

Boundary conditions and mechanical protocols. The modular structure of MUSICO permits variety of mechanical protocols following the exact experimental conditions. The mechanical loadings include prescribed length or force on muscle fiber ends. Multisarcomere structure (in series), extensibility of myosin and actin filaments and effect of other structural elements (e.g. elasticity of tendons) interconnect non-affine deformations from the fiber ends to actual strain dependent molecular interactions.

Nonlinear mechanical behavior of extensible filaments and crossbridges is addressed using an iterative finite element scheme. The platform is both adaptable, for different muscle systems, scalable and parallelizable in order to address larger and larger ensembles as available computing power permits. MUSICO, in its present form, can address a wide range of physiological conditions including fast transients (T1-T2), transients from isometric to isotonic shortening, as well as oscillatory contraction. While other spatially explicit models exist they have been limited in scope to only a few physiological states and have only considered structures up to a half sarcomere, MUSICO can simulate multiple sarcomeres in series (up to 10, so far) with each sarcomere consisting of 500 to 1000 myosin filaments in parallel.

MUSICO outputs Include traces of force, length, muscle stiffness, ATPase and X-ray diffraction patterns. For detailed analysis specifically prescribed outputs can include the distribution of crossbridge forces at prescribed instants, evolution of forces in myosin and actin filaments and their variations, the distribution of monomer spacing along the filaments etc.

The correlation of biochemistry and mechanical data on the cross bridge cycle.

Our approach provides structurally-based, rather than empirical, theoretical constraints for strain-dependent actomyosin interaction kinetics. In striated muscle, including cardiac muscle, myosin and actin filaments are packed into a regular three-dimensional lattice. Each myosin filament is decorated with crowns of myosin dimers, spaced by ~14.3 nm along the filament. Crown orientations are repeated every 42.9 nm, i.e. every third crown has the same orientation as the initial. The actin filaments form a double-helix, associated with its regulatory proteins tropomyosin and troponin with binding sites by ~5.5 nm on each strand, with a half-period of ~35.75 nm in the relaxed state. The difference in periodicities between actin binding sites (35.75 nm) and myosin crowns (42.9 nm) creates a vernier of longitudinal head-site spacings which control strain dependent binding of myosin to actin filament. The 3D geometry of myosin head binding domains and actin sites in a sarcomere requires both longitudinal position matching and angular matching in the azimuthal plane. A myosin head and closest actin site form the most probable pair of these molecules which can create a crossbridge interconnecting actin and myosin filaments. Thus, structural changes along the reaction pathway must be responsible for the spatial strain dependence of the force-generating and product-release steps in actomyosin in the 3D sarcomere lattice. Our model therefore, incorporates of azimuthal weight factors and precise (nonlinear) elasticity of crossbridge in order to construct realistic energy landscapes.

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