(A) Graphical representation of the finite-element muscle fiber array. (B) Free body diagram of two sarcomeres linked at a Z-disk. Viscoelastic sarcomere elements are linked longitudinally at Z-disk nodes to form myofibrils and laterally by desmin elastic filaments (blue) to form a two-dimensional fiber. ECM (and sarcolemma composite) elements form the upper and lower bounds of the array, whereas tendon elements define each end. The location of each element is indexed by its row (i) and column (j). The array continues to extend in the direction of increasing i and j until it meets another ECM/tendon junction. The contractile element (CE), the series elastic element (SE), the parallel viscoelastic element (PVE), the sarcomere length (SL), and the series spring extension (E) are labeled in the bold sarcomere. Forces involved in the calculation of the nodal equations of motion are illustrated by color-coded arrows (red, active; blue, elastic; and green, viscous). The parameters used in the calculation of these forces are labeled near their associated arrows, with forces listed in capital letters and constants in lower case. Dotted arrows and associated variables label the directions of positive nodal displacement from the initial position. Parameters and nodal displacements are also indexed with subscripts by their location in the finite-element array in the (row, column) format. Sign conventions are set assuming that x_i−1 < x_i < x_i+1. Variables x_i,j and y_i,j are the displacements at the x and y nodes, respectively. F_E are the forces external to the sarcomeres pictured and F_i,j is the contractile force in the CE of sarcomere (i,j). PT_i,j is the passive tension developed in the PVE and c_i,j is the damping constant of the PVE of the same sarcomere. Constant s_i,j is the stiffness of the SE and k_i,j is the stiffness of desmin.

Passive stress of myofibril elements and ECM elements, and the average passive stress in the simulated fiber superimposed on the sarcomere length-tension diagram. Myofibril passive stress, ECM passive stress, and the sarcomere length-tension relationship were estimated from work by others (6,13,15, 25–27). The fiber average passive stress is calculated for a simulated fiber with 10 myofibrils laterally by summing the force contribution of each myofibril and the two ECM elements and dividing by the total cross-sectional area.

Sarcomere length distributions across the fiber length and depiction of initial passive tension (inset). The black trace is the sarcomere length distribution generated by Eq. 8, with a mean sarcomere length of 2.2 μm and a length range of 0.2 μm. The gray trace shows what happens to the distribution after passive stretch to a mean sarcomere length of 3 μm. The variation in sarcomere length is a result of the random variation in the passive tension coefficient, λ. (Inset) Passive tension curves for sarcomeres 1 and 100, where the curve for sarcomere 1 has been multiplied by a larger coefficient so that the passive tensions are equal for the two sarcomeres at their initial lengths (dots on inset).

(A) Mean Z-disk stagger as a function of sarcomere length over the course of a passive stretch. (B) Measurements of stagger at every 10th Z-disk over the length of the fiber at a sarcomere length of 5 μm. Data points are the mean value from 10 runs in each group, with a different random variation in the passive tension constant used for each run, and error bars are the standard deviations. Desmin null fibers have a significantly higher mean Z-disk stagger than wild-type at all sarcomere lengths >3 μm and at all Z-disks except for the fiber ends, which are constrained. Some error bars are small and are obscured by the data marker.

Length traces over time of a representative run for selected sarcomeres from wild-type (A) and desmin null (B) fibers and the overall normalized force traces (C) during simulated fiber fixed-end contraction. Solid lines correspond to wild-type simulations and dotted lines to desmin null. Sarcomere numbers are increasing along the length of the fiber, and each number is shown adjacent to the corresponding trace on the right side of the plot. This plot shows that end sarcomeres shorten significantly over the contraction period, whereas center sarcomeres lengthen more slowly and continuously. Note that end sarcomeres in the desmin null traces (B) are shorter than in the wild-type (A), and their corresponding force is lower (C). Compare with simulations reported previously (Figs. 4 A and 5 in Morgan et al. (9)).

(A) Z-disk stagger as a function of the time course of simulated fixed-end contraction. (B) Measurements of stagger at every 10th Z-disk over the length of the fiber at 2.5 s. Mean Z-disk stagger is significantly higher for desmin null fibers than for wild-type at all time points and increases nonlinearly as a function of time. Z-disk stagger and hence differences in stagger between wild-type and desmin null fibers are small at the fiber ends due to the abrupt force drop modeled in the contractile element, with shortening beyond 1.68 μm sarcomere length and the limited opportunity for sarcomere length variations to add up to large staggers due to the proximity to the tendon. Z-disk staggers are on the same order as those seen during passive stretch.

Plot of the maximum fixed-end stress production during simulated fixed-end contraction of fibers with varying concentrations and localizations of desmin. Stress production is greater when desmin is preferentially localized to the fiber membrane (green) than when it is distributed randomly (blue) or preferentially localized away from the fiber membrane (black). Each data point represents the mean of 10 simulations under its color-coded condition, and error bars are standard deviations.