Figure 1.
Schematic depiction of sarcomeric organization in myofibrils.
Actin filaments (blue and red) are grafted at their plus-ends in an -actinin rich crosslinking band, termed the Z-band (green). The repetitive units spanning from one Z-band to the next are referred to as sarcomeres and measure
in length. The myosin band (magenta) is traditionally called A-band, while the myosin-free part of the actin band is called I-band. Numerous auxiliary proteins ensure structural integrity and tune elastic properties.
Figure 2.
Actin cluster formation and coalescence.
A. Our computational model of sarcomeric pattern formation considers a one-dimensional bundle of parallel actin filaments, which undergo treadmilling, i.e. filaments polymerize at their plus-ends and depolymerize at their minus-ends resulting in a net motion of the plus-end with respect to the individual monomers. Plus-end tracking crosslinkers (green) can permanently attach to the plus-ends of actin filaments (blue and red, indicating filament polarity), while still allowing for polymerization at filament plus-ends. B. Plus-end tracking crosslinking results in the formation and coalescence of actin clusters as reflected by a reduction in the number of actin clusters (single actin filaments are counted as one cluster). If there is no friction between sliding filaments (), all actin clusters eventually coalesce into a small number of very large clusters (blue, mean
s.e.,
). Time is measured in units of actin length divided by treadmilling speed,
. In the presence of inter-filament friction (
), however, actin clusters above a critical size effectively repel each other, resulting in a kinetically stabilized configuration with a finite number of actin clusters (magenta). To the right, example kymographs of actin cluster coalescence are shown for the cases without friction and with friction, respectively. A small imbalance in the number of filaments treadmilling either to the right or to the left within the final striated bundle causes a slow motion of the entire bundle as a whole as is reflected by the tilted cluster trajectories. Using static instead of periodic boundary conditions impedes this motion, see SI text S1. The color scheme encodes filament number in actin clusters as shown in the color bar.
Figure 3.
Sarcomeric ordering in the presence of myosin.
A. Simulation snap-shots showing the emergence of sarcomeric order in an acto-myosin bundle ( single actin filaments: blue and red, myosin filaments: magenta, plus-end crosslinker connecting actin filament plus ends belonging to one cluster : green). Actin filaments can interact if their projections on the bundle axis overlap. Additionally, bipolar myosin filaments (magenta) dynamically attach to actin filaments in a polarity-specific manner, thus acting as a second set of active crosslinkers. Different vertical positions of the filaments are indicated solely for visualization purposes. Initially, filament positions are random (). After a transient period during which clusters of crosslinked actin filaments form and coalesce (
), a stable configuration is established characterized by a periodic pattern of actin clusters interspersed by bands of aligned myosin (
). To characterize sarcomeric order in these simulated bundles, we compute the structure factor
as defined in the main text (blue curves in lower panel, simulation time
, respectively). The height of the principal Bragg peak (red circle) defines the sarcomeric order parameter
. The active myosin force that tends to oppose sarcomeric ordering was chosen as
, measured in units of
. An animated version of this simulation can be found as Video S1 available online as Supplementary Information. B. Illustration of the ‘actin conveyer belt’ mechanism: Actin filaments that are grafted at their plus-end by a processive crosslinker have to polymerize against the crosslinker (that acts as an obstacle) and are pushed backwards in a form of local retrograde flow. Myosin filaments interacting with these treadmilling actin filaments are transported away from the cluster center provided that the actin treadmilling speed exceeds the active myosin walking speed. C. Myosin filaments that are attached to actin filaments from two neighboring clusters serve as an active crosslinker and mediate repulsive forces between the two clusters due to the difference in the actin polymerization forces and the myosin active forces, see also SI text S1. D. Myosin active force slows-down sarcomeric ordering: The inset shows the time-course of the sarcomeric order parameter
(blue,mean
s.e.,
) for
, together with a fit of simulation results to an exponential saturation curve
(red) that allows us to extract a time-scale
of sarcomeric ordering. The main plot shows this time-scale
as a function of myosin force
;
diverges as
approaches a critical value
. For myosin forces that are larger the critical value
, sarcomeric order is not established. Instead, myosin forces facilitate the coalescence of crosslinked actin clusters into a small number of very large clusters (not shown), similar to the case shown in figure 2B without friction.
Figure 4.
Sarcomeric order despite actin filament length variability.
In a modified version of the simulations shown in figure 3, the lengths of individual actin filament were chosen from a unimodular length distribution, see main text. Example length distributions are shown for three values of the length variability parameter . Sarcomeric order also evolved in simulated acto-myosin bundles with a distribution of filament lengths, but with a reduced sarcomeric order parameter and increased sarcomere spacing at steady-state (mean
s.e.).
Figure 5.
Myosin order despite actin turnover.
We devised a minimal model of actin filament turnover, see main text. For simulations as in figure 3, but with actin turnover, the sarcomeric order parameter was found to decrease as a function of actin filament turnover rate (blue curve) as actin turnover impedes the formation of large actin clusters (blue, means.e.,
). Surprisingly, an analogously defined order parameter for myosin positions attains significant values even for considerable actin turnover rates. A simulation snap-shot at
is shown to the right for actin turnover rate
(in units of
).
Figure 6.
Actin filament length control by severing.
A. Filament severing provides a simple physical mechanism for actin filament length control, see main text. In an idealized scenario, an actin filament (blue) binds a severing agent (scissors) with a rate that is proportional to its length
at a random position. The filament is then cut at the binding position, and its minus-end facing fragment is subsequently depolymerized. B. Actin filament severing results in a unimodular filament length distribution at steady state, see histrogram (gray) and analytical expression (red, see SI text S1). For the severing rate used,
, mean filament length
, and filament length variability parameter,
. C. Simulation of an acto-myosin bundle as in figure 3, but with actin filament severing as described in panel A. Shown is a snap-shot of the simulations at time
(actin filaments: blue and red; myosin: magenta; end-tracking crosslinker: green), as well as the averaged structure factor (black curve, gray region indicates mean
s.e.,
).