A novel three-filament model of force generation in eccentric contraction of skeletal muscles.

We propose and examine a three filament model of skeletal muscle force generation, thereby extending classical cross-bridge models by involving titin-actin interaction upon active force production. In regions with optimal actin-myosin overlap, the model does not alter energy and force predictions of cross-bridge models for isometric contractions. However, in contrast to cross-bridge models, the three filament model accurately predicts history-dependent force generation in half sarcomeres for eccentric and concentric contractions, and predicts the activation-dependent forces for stretches beyond actin-myosin filament overlap

Computing Average Passive Forces in Sarcomeres in Length-Ramp Simulations.

Passive forces in sarcomeres are mainly related to the giant protein titin. Titin's extensible region consists of spring-like elements acting in series. In skeletal muscles these elements are the PEVK segment, two distinct immunoglobulin (Ig) domain regions (proximal and distal), and a N2A portion. While distal Ig domains are thought to form inextensible end filaments in intact sarcomeres, proximal Ig domains unfold in a force- and time-dependent manner. In length-ramp experiments of single titin strands, sequential unfolding of Ig domains leads to a typical saw-tooth pattern in force-elongation curves which can be simulated by Monte Carlo simulations. In sarcomeres, where more than a thousand titin strands are arranged in parallel, numerous Monte Carlo simulations are required to estimate the resultant force of all titin filaments based on the non-uniform titin elongations. To simplify calculations, the stochastic model of passive forces is often replaced by linear or non-linear deterministic and phenomenological functions. However, new theories of muscle contraction are based on the hypothesized binding of titin to the actin filament upon activation, and thereby on a prominent role of the structural properties of titin. Therefore, these theories necessitate a detailed analysis of titin forces in length-ramp experiments. In our study we present a simple and efficient alternative to Monte Carlo simulations. Based on a structural titin model, we calculate the exact probability distributions of unfolded Ig domains under length-ramp conditions needed for rigorous analysis of expected forces, distribution of unfolding forces, etc. Due to the generality of our model, the approach is applicable to a wide range of stochastic protein unfolding problems

Mean ± standard deviation of stress versus average sarcomere length of myofibrils.

<p>Myofibrils were stretched in a low-calcium solution (passive stretch, open circles) and high-calcium solution from an initial sarcomere length of 2.4μm (active stretch, black circles) and 3.4μm (active stretch from 3.4μm/sarcomere, grey squares). Based on the cross-bridge model, forces beyond actin-myosin filament overlap (sarcomere length > 4μm; grey vertical bar) in actively stretched myofibrils are predicted to coincide with forces in passively stretched myofibrils. However, forces in actively compared to passively stretched myofibrils are three to four times higher when the stretch starts at an initial sarcomere length of 2.4μm, and around twice as high when the stretch starts at an initial sarcomere length of 3.4μm. Careful experimental and theoretical testing and analysis of all individual sarcomere lengths revealed that these results cannot be explained by the development sarcomere length non-uniformities. A detailed description of the experiments is given elsewhere [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0117634#pone.0117634.ref002" target="_blank">2</a>].</p

Force-elongation predictions of sarcomeres stretched beyond actin-myosin filament overlap.

<p>When a sarcomere is activated at 3.4μm and then stretched actively beyond actin-myosin overlap, its force will exceed the purely passive forces, but will not reach the high forces of sarcomeres stretched actively from optimal length.</p

Comparison between the two methods.

<p><b>A</b> Force-elongation curves of the Monte Carlo simulation. The red line indicates the mean value of force at a given half sarcomere length. The resulting curve is similar to the exact solution <b>B</b>. The small insert shows the difference between the two methods. The error of the Monte Carlo approximation rises with the first unfolding events but stays well within a 2pN range. The probability function and the normalized histograms (bin size based on the Freedman-Diaconis rule [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004904#pcbi.1004904.ref044" target="_blank">44</a>]) show similarities in shape.</p

Repeated hysteresis loops.

<p><b>A</b> Monte Carlo simulations and <b>B</b> exact solution of hysteresis loops based on one Ig cluster of the following setup: a half sarcomere was subject to two subsequent stretch-shortening cycles. After a resting period of 30s another stretch-shortening cycle was performed. We observed that while hysteresis was significantly reduced in the second cycle it fully recovered in the third cycle due to refolding of Ig domains in the resting period [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004904#pcbi.1004904.ref005" target="_blank">5</a>]. The effect of repeated stretch-shortening cycles on the unloading energy remains negligible.</p

Forces at the first unfolding event.

<p><b>A</b> Histogram of forces at the first unfolding event (bin size based on the Freedman-Diaconis rule [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004904#pcbi.1004904.ref044" target="_blank">44</a>]) of 200 Monte Carlo simulation reveal a deviation (in terms of most likely unfolding forces) from the corresponding exact probability <b>B</b>.</p

Sketch of titin strands in a half sarcomere.

<p>While a single titin strand spans the whole half sarcomere only a part of titin located in the half I-band is extensible.</p

Force versus average sarcomere length of a myofibril first stretched in a low-calcium solution and then shortened again.

<p>The red line corresponds to region (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0117634#pone.0117634.e001" target="_blank">1</a>) where no unfolding of IG domains took place; the green line to region (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0117634#pone.0117634.e005" target="_blank">4</a>) where no further folding-unfolding of IG domains took place.</p