Myosin order despite actin turnover.

<p>We devised a minimal model of actin filament turnover, see main text. For simulations as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002544#pcbi-1002544-g003" target="_blank">figure 3</a>, 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 ).</p

Sarcomeric ordering in the presence of myosin.

<p><b>A.</b> 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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002544#pcbi.1002544.s002" target="_blank">Video S1</a> available online as Supplementary Information. <b>B.</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. <b>C.</b> 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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002544#pcbi.1002544.s001" target="_blank">text S1</a>. <b>D.</b> Myosin active force slows-down sarcomeric ordering: The inset shows the time-course of the sarcomeric order parameter (blue,means.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 <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002544#pcbi-1002544-g002" target="_blank">figure 2B</a> without friction.</p

Schematic depiction of sarcomeric organization in myofibrils.

<p>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.</p

Distinguishing head morphologies of four different flatworm species.

<p><b>A.</b> Application of our method to parametrize head morphology of four different flatworm species. For each species, time-lapse sequences of different worms were recorded as two independent runs of duration frames. The head is defined as most anterior of the worm body. Radial distances are computed with respect to the midpoint of the head (red dot at of the worm length from the tip of the head). <b>B.</b> By applying PCA to this multi-species data set, we obtain two shape modes, which together account for of the shape variability. Deformations of the mean shape with respect to the the two modes are shown (black: mean shape, red: superposition of mean shape and first mode with and second mode with , respectively). We represent head morphology of the four species in a combined shape space of these two modes. Average head shapes for each species are indicated by crosses, with ellipses of variance including (dark color) and (light color) of motility-associated shape variability, respectively.</p

The mathematics behind principal component analysis (PCA).

<p><b>A.</b> For illustration, we start with a measurement matrix featuring the beat of a sperm flagellum with measurement (rows) and tangent angles at equidistant positions along the flagellar centerline (columns). Subtracting the mean defines the centered data matrix . The mathematical technique of singular value decomposition factors the data matrix into a product of a unitary matrix , a “diagonal” matrix that has non-zero entries only along its diagonal, and a unitary matrix . Singular value decomposition may be regarded as a generalization of the usual eigensystem decomposition of symmetric square matrices to non-square matrices. A unitary matrix U generalizes the concept of a rotation matrix to <i>n</i>-dimensional space.;it is defined by being equal to the identity matrix. <i>Second row:</i> A restriction to the top- singular values defines sub-matrices of , , of dimensions , , , respectively, whose product represents a useful approximation of the full factorization that reduces feature dimensions to only shape modes. <b>B.</b> The feature-feature covariance matrix is defined in terms of the centered data matrix . It can be written as a product of a diagonal matrix , whose diagonal features the eigenvalues of and a unitary -matrix V whose columns correspond to the respective (left) eigenvectors of . This matrix V is exactly the same as previously encountered in the singular value decomposition of . <b>C.</b> Similarly, the measurement-measurement covariance matrix , known as the Gram matrix, can be decomposed using a diagonal matrix and a unitary matrix . Importantly, the rows of V comprise just the shape modes of the data matrix as defined by linear PCA, while the columns of the matrix yield the corresponding shape scores.</p

Principal component analysis is used across different disciplines, giving rise to a diverse terminology, which is summarized here.

<p>Principal component analysis is used across different disciplines, giving rise to a diverse terminology, which is summarized here.</p

Chemotactic success with decision making.

<p>Success probability <i>P</i>(<i>R</i>₀) for the optimal decision strategy, resulting from switching between ‘low-gain’ and ‘high-gain’ steering, as function of initial distance <i>R</i>₀ to the egg for the case of noise-free concentration measurements (A), and physiological levels of sensing noise (B) (red squares). For comparison, success probabilities for strategies without decision making are shown (circles). (C,D) Optimal decision strategies for the cases shown in panel A and B. Greyscale represents prediction frequency of ‘high-gain’ steering, using a cohort of MDPs obtained by bootstrapping, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.s001" target="_blank">S1 Appendix</a> for details. Arrows and dashed lines indicate zone boundaries as introduced in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.g002" target="_blank">Fig 2</a>. (E,F) Spatial sensitivity analysis of optimal strategies: Shown is the change in chemotactic range as function of cut-off distance <i>R</i>_<i>c</i> for hybrid strategies that employ the optimal strategy for <i>R</i> < <i>R</i>_<i>c</i>, and either ‘low-gain’ steering (white circles) or ‘high-gain’ steering (black circles) else. Positive values indicate a benefit of decision making at the respective distance to the egg. Parameters, see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006109#pcbi.1006109.s001" target="_blank">S1 Appendix</a>.</p

Three shape modes characterize projected flatworm body shape dynamics.

<p><b>A.</b> Our custom-made MATLAB software tracks worms in movies and extracts worm boundary outline (red) and centerline (blue). <b>B.</b> The radial distance between the boundary points and midpoint of the centerline (, red dot) is calculated as a parameterization of worm shape. We normalize the radial distance profile of each worm by the mean radius . <b>C.</b> The second symmetry axis (dotted line) of the covariance matrix corresponds to statistically symmetric behavior of the worm with respect to its midline. <b>D.</b> The three shape modes with the largest eigenvalues account for 94% of the shape variations. The first shape mode characterizes bending of the worm and alone accounts for 61% of the observed shape variance. On the top, we show its normalized radial profile on the left as well as the boundary outline corresponding to the superposition of the mean worm shape and this first shape mode (solid red: , dashed red: , black: mean shape with ). The second shape mode describe lateral thinning (), while the third shape mode corresponds unlike deformations of head and tail (), giving the worm a wedge-shaped appearance. <b>E.</b> The first shape mode with score describing worm bending strongly correlates with the instantaneous turning rate of worm midpoint trajectories. <b>F.</b> We manually selected 30 movies where worms clearly show inch-worming and 50 movies with no inch-worming behavior. The variance of score and increases for the inch-worming worms. <b>G.</b> The autocorrelation of mode and the crosscorrelation between mode and mode reveals an inch-worming frequency of approximately , hinting at generic behavioral patterns.</p