Origin of crossbridge stiffness. 

 

Chemo-mechanical coupling within the actomyosin cycle is strongly affected by the strain-dependence of the biochemical transitions that comprise the cycle. The physical basis for these dependencies lies in the compliance of the S2, crossbridge “neck” region and lever arm. Current estimates of total cross-bridge stiffness vary considerably, thus making it difficult to define a precise model of single fiber force development during contraction and its transients. For example, the estimate of cross-bridge stiffness derived from early tension recovery recordings or subsequent single molecule experiments may in fact be greater than predicted if the compliance due to thick and thin filament extensibility is also considered.  This correction brings cross-bridge stiffness to the range estimated from maximum thermodynamic efficiency, which is similar to that obtained using optical tweezers.The most recent experiments report values of myosin stiffness (1.5 to 3.2 pN/nm) that are in fact larger than previous estimates. Closer examination of the binding process is necessary in order to permit an estimate of a reasonable number of attached heads in a 3D sarcomere lattice.  After binding, a working stroke is energetically favored when the initial strain is negative (i.e. crossbridge is in compression), but the energy cost of a 10 nm working stroke after thermal binding will still be prohibitively high unless the working stroke is accompanied by a large drop in chemical energy. In this regard, length-step experiments show that the working stroke can be reversed by stretch and that derived rate constants for the forward and backward strokes provide estimates of the intrinsic energy drop on the order of 4 during the working stroke. Considering these uncertainties, independent estimates of cross-bridge stiffness are needed to determine the contributions of S2, the lever arm, the unstructured segment between lever arm and the S2 subdomain termed the “neck region,” the myosin motor domain, and the actin-myosin connection to overall cross-bridge compliance.

Nonlinear elasticity of a crossbridge in sarcomere lattice. 

 

M. Kaya and H. Higuchi measured the stiffness of single skeletal myosin using optical trapping and fluorescence imaging techniques. They reported that “stiffness is dramatically lower for negatively compared to positively strained myosins, consistent with buckling of myosin’s subfragment 2 (S2) rod domain”. Their measurements confirmed the nonlinearity of skeletal myosin molecule stiffness during compression, due to the bending and buckling of S2. While the large fall of stiffness in compression is correctly interpreted, it is puzzling from where is the large nonlinear drop in stiffness in tension coming from? In fact, the largest fall in reported stiffness is from 2.5-2.9 pN/nm at tension of 2 pN to stiffness of 0.75 pN/nm at zero tension. In this range there is no compression and therefore no buckling. This large nonlinear drop in stiffness is not observed in fibers, at least not in this degree, during, for example rapid drop in length.

Comparison of stiffness- and force-displacement relationships between original Kaya and Higuci’s data and the NFE simulations. The parameters used in simulations are: axial rigidity of lever arm        = 1800 pN, and of S2       = 3600 pN; bending rigidity of lever arm         = 1200  pN∙nm2 and of         = 600 pN∙nm2; stiffness of S1-S2 joint is 100 pN∙nm and of myosin filament S2 joint is 50 pN∙nm. Length of lever arm is 10.5 nm and of S2 is 60 nm. Actin-myosin filament lattice size is       = 36.5 nm. The predictions (red lines) excellently fit observation after shift (green triangles up). In force length curve, blue dashed line represents original measurements while black line and open circles represent the data corrected for actin and myosin compliances.

To resolve this problem we performed nonlinear finite element (NFE) analysis to estimate the crossbridge stiffness under a range of tensile and compressive forces in the context of the 3-D sarcomere lattice. The elasticities of the individual crossbridge components used in simulations are calculated from the known atomic structures using molecular dynamic (MD) simulations (CHARMM). Our simulations confirmed the large fall of stiffness during compression (see Figure), but required shift in the nonlinear stiffness-displacement relationship. The simulations of the crossbridge stiffness, showed that the Kaya and Higuchi’s data can be almost perfectly fitted with our simulations if the zero of their force-displacement relationship is shifted downward for 3.35 pN and leftward for 1 nm. In our data (see Figure) the dramatic drop in stiffness is almost exclusively in compression. Furthermore, the shape and magnitude of both, force-displacement and stiffness-displacement relationships, closely follow the reported data after the shift. We believe that only small nonlinear decrease in stiffness is possible at low tension. Establishing correct nonlinear relation of crossbridge stiffness is exceptionally important in understanding strain dependence of mechanochemical transitions in actin-myosin cycle.