Fluid mechanics of sarcomeres as porous media

Figures

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  Fig. 1. Physical and mathematical modelling of sarcomere. (A) A muscle fibre containing many myofibrils.10 (B) A single myofibril, containing many aligned sarcomeres. Each Z disk defines the boundary from one sarcomere to the next.10 (C) Electron micrograph of a single sarcomere, showing the defining banding patterns of striated muscle;11 regions of different intensities correspond to distinct regions in the sarcomere, which are labelled as in D. (D) Simplified structure of a single sarcomere, with thin actin filaments (blue), thick myosin filaments (red), and Z and M disks; the three regions are defined by the presence of the thick and/or thin filaments: regions 1 (thick only), 2 (overlap) and 3 (thin only). (E) Darcy model representation of the sarcomere as an anisotropic porous medium (with the three regions highlighted in different colours), showing radius R and overall sarcomere length 2L. (F) Blow-up of top-right quarter of the Darcy model representation, with non-dimensionalisations giving radius and half-length of 1, and boundary conditions as indicated.

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  Fig. 2. (A) Sarcomere cross-section in the overlap region (i.e. region 2 in Fig. 1D), here illustrated for a sarcomere of hypothetical radius 200 nm, with a 30° sector of symmetry indicated. (B) 30° sector of cross-section, as indicated in (A). (C)–(E) Hexagonal arrangement of filaments in the overlap region. Packing ratios (number of thin filaments per thick half-filament) of 2 (C), 3 (D) and 5 (E) are shown, with 2 being typical for vertebrates and 3 and 5 often being found in invertebrates.9,42,43 Fundamental triangles, each being 1/12 of a hexagon, are shown at the bottom. (F)–(H) Fundamental triangles, highlighted in (C)–(E), used to compute the permeabilities and the traction parameter β.

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  Fig. 3. Comparisons of fluid flow between the Darcy and FEM models, for a sarcomere at 50% contraction with a packing ratio of 5 in the absence of an M disk. (A) and (B) Heat map of fluid flux with arrows indicating direction for Darcy (A) and FEM (B) models. Note that in A, the maximum flux magnitude (found at the radial boundary, at the interface between regions 1 and 2) is approximately 1600 nm s−1, whereas the colour scale has a maximum value of 1100 nm s−1; this is to improve readability of A and B, and less than 0.1% of the domain surpasses 1100 nm s−1. (C) Plot of radial flux against z for the Darcy (red) and FEM (blue) models, with r = 100, 200, 300, 400, 500 nm from bottom to top. Radial fluxes have been individually rescaled to have the same maximum value of radial Darcy flux, to improve readability.

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  Fig. 4. Comparisons of fluid flow between the Darcy and FEM models, for a sarcomere at 50% contraction with a packing ratio of 5 in the absence of an M disk. Plot of axial flux against r for the Darcy (red) and FEM (blue) models, with z = 100, 300, 500, 700, 900, 1100, 1300 nm from bottom to top. Axial fluxes have been individually rescaled to have the same maximum value of axial Darcy flux, to improve readability.

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  Fig. 5. Lagrangian deformation of marked fluid particles throughout a sarcomere contraction, shown for contractions of α = 5% (A), 25% (B), 50% (C), 75% (D) and 95% (E). Dashed black lines indicate interfaces between the three regions.

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  Fig. 6. Radial fluid flux in a sarcomere with a packing ratio of 3 (cross-section of overlap region illustrated on top left) in the absence of an M disk at various stages of contraction, α = 5% (A), 25% (B), 50% (C), 75% (D), 95% (E), showing the Darcy model (red) against the FEM model (blue). Each figure shows five pairs of data sets, at r = 100, 200, 300, 400, 500 nm from bottom to top.

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  Fig. 7. Radial fluid flux in a sarcomere with a packing ratio of 2 (cross-section of overlap region illustrated on top left) in the presence of an M disk at various stages of contraction, α = 5% (A), 25% (B), 50% (C), 75% (D), 95% (E), showing the Darcy model (red) against the FEM model (blue). Each figure shows five pairs of data sets, at r = 100, 200, 300, 400, 500 nm from bottom to top.

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  Fig. 8. Triangular domain, 1/12 of a periodic hexagonal cell, on which the axial and lateral flow profiles are calculated, shown here for the overlap region with a packing ratio of 5 (see Fig. 2). The triangle's original height of is rescaled (using a different non-dimensionalisation to that used for the Darcy model) to 1. Polar coordinates (r,θ), distinct from those used for the Darcy model, are shown, with rescaled thick and thin filament radii Rthick and Rthin indicated.

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  Fig. 9. ATP concentration over the course of a contraction, in a human cardiac sarcomere with no advection (A) and advection by the fluid flow (B). Snapshots of ATP concentration are shown at low (i), intermediate (ii) and high (iii) levels of human cardiac sarcomere contraction. Horizontal axis represents the dimensionless axial coordinate of the sarcomere, and the vertical axis represents the dimensionless radial coordinate. Dashed lines indicate transitions between the key regions of the sarcomere, and reaction occurs in the middle region only.

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  Fig. 10. Total dimensionless radial efflux in regions 1 (A), 2 (B) and 3(C) due to variations of k‖(3) and k⊥(3) by factors between 0.3 and 2, compared to their values in Table 2. The porosity Φ(3) in the thin region is fixed at its established value used in previous examples in the main text.