X-Ray Diffraction: Estimation of Forces Acting on Structures within Living Cells

Small-angle X-ray diffraction of contracting muscle during various mechanical protocols provides a view of sarcomere structure averaged over the area illuminated by the X-ray beam. Many important details of muscle contraction, such as crossbridge attachment and detachment rates are likely to depend on the local structures and forces on these structures. We have developed a methodology for predicting X-ray diffraction patterns with stepwise increases of strain along actin filaments in the hexagonal sarcomere lattice. Using PDB data for the crystal structures of G-actin we reconstructed the geometry of actin filaments deformed under stress generated by crossbridges. This stress varies along the filament length, so these deformations also vary with length. These fiber deformations may be determined by Monte Carlo calculations using the computational platform, MUSICO, along with the force and length changes seen during classical mechanical protocols. Using the predicted changes in spacings of actin filaments due to crossbridge forces we predicted how the X-ray diffraction patterns vary with force and along length during mechanical transients.

The MUSICO computational platform interconnects data collected at molecular scale, including modification of structural and kinetics properties by specified mutation(s) and muscle fibre or muscle tissue mechanical and metabolic data. The X-ray diffraction data verify predicted (by MUSICO) collective movement of sarcomeric proteins during contraction.

To date, modeling the behavior of the isolated protein and its effect on the whole sarcomere has been done only for various approximate models. Using MUSICO platform it can be done with a correct geometric model and with built-in elasticity and mechanochemical transitions. Comparing MUSICO predictions to simultaneous structural measurements at nano-scale by X-ray diffraction and recordings of muscle response at cell or tissue level provides unique validation of the multiscale model at two length scales. 

Generated X-ray diffraction patterns from MUSICO simulations during isometric contraction in multiple sarcomere 3D lattices with extensible filaments show smeared layer lines, caused by different pitches of helices - reflecting nonuniform deformation along actin filaments. Calculated meridional X-ray diffraction peak profiles from deformed helices are skewed and closely resemble the observed profiles by Huxley et al. 1994 and Wakabayashi et al., 1994, 2006. Simulated X-ray patterns show that the 3rd actin meridional reflection best describes degree of non-uniformity of actin filament reflecting the changes in molecular spacings along the filaments during force development and relaxation. This methodology provides unique tool for measurement of force acting on a single myosin or actin filament and therefore more precise assessment of crossbridge forces.

X-ray diagram for frog sartorius muscle at rest. (right side) and during isometric contraction (left side). Enlarged picture of 27.3 Å actin and 28.6 Å myosin meridional reflections, at rest (Relaxed) and during isometric contraction (Contracted).

MUSICO predicted X-ray diffraction patterns of actin monomer spacing in Relaxed and Contracted Muscle. Enlarged pictures of 27.3 Å, 13.65 Å and 9.10 Å actin meridional reflections, at rest (Relaxed) and during isometric skeletal muscle contraction (Contracted).

Simulations of thin filament forces and compliance. Predicted X-ray diffraction patterns of fully developed isometric force muscle fibers are compared to experimental X-ray diffraction measurements of Huxley et al., 1994 in order to estimate the average force in actin filaments. Using peak shape analysis of the meridional reflections, one can assess the variation of filament deformation, and hence force, along the length of the filaments. Stochastic spatial 3D binding of myosin to actin results in random stepwise increase of force, thus creating random stepwise increase in subunit spacings. The axial profiles of the first actin meridional reflection at ~27.3 Å for different actin filaments in the same muscle show only subtle differences but, however, the profiles of the second actin meridional reflections at ~13.65 Å show notable differences and variations in shape. By comparing predicted X-ray diffraction patterns of 27.3 Å and 13.65 Å peaks to experimental X-ray diffraction measurements of Huxley et al., 1994 we constructed the relationship between the spacing change going from the relaxed to the contracted state (Del H), and the forces in actin filaments in 5 different experiments.

Estimated mean actin force and SD from 5 experiments. From inverse spacing peaks at of ~27.3 Å and ~13.65 Å  we calculated change in spacing between contracted and relaxed spates (Del H). Calculated MUSICO patterns are fitted into the experimental peaks ~27.3 Å and ~13.65 Å to match the position of the peaks (in inverse space) and their width. Good fit provided the mean and SD of forces in actin filaments (close to Z-lane). The estimated mean tension is calculated form the mean actin force per myosin filament assuming                 nm and 70 % of cross-section occupied by myofilaments.

Good matches were obtained of the calculated to the experimental X-ray peak widths allowing estimation of the mean forces and their SD in the actin filaments close to the Z-line (Table). The variation of the forces between actin filaments is caused by stochastic nature if myosin binding in explicit sarcomere lattice.

Relationship between forces in actin filaments and inverse spacing of contracted fiber. The histograms of actin filament forces (close to Z line) calculated from MUSOCO.  Fitting predicted X-ray diffraction patterns of 27.3 Å and 13.65 Å peaks we constructed relationship between inverse spacing in contracted state, i.e. spacing change from relaxed to contracted state, and the forces in actin filaments. Using the same procedure we estimated mean actin force from 5 experiments and data are presented in the Table.

In summary, the developed the methodology enables direct measurement of forces in myofilaments in living myosin fibers. Furthermore this methodology provides unique tool for measurement of force distributions acting along single myosin or actin filament and therefore more precise assessment of crossbridge forces. The detailed analysis showed that the     actin meridional reflection best describes the position of the peak and width of the     meridional reflection includes both the degree of non-uniformity of spacing along each actin filament and the variation of forces in different actin filaments. MUSICO simulations can separate these two effects on the width and shape of     and     meridional reflection showing large variation in forces actin on different filaments.

Our approach can be extended to any muscle system, and it could ultimately provide an interpretive framework for studying the mechanisms of inherited or acquired diseases. By extracting maximum information from the X-ray patterns, in combination with the physiological data this approach provides a template to test hypotheses concerning crossbridge and regulatory protein action in working muscle.

The near future applications may include MUSICO analysis of meridional X-ray diffraction patterns from actin filaments in WT and               mouse soleus muscle taking into account the contributions of nebulin as well as other accessory proteins. In addition, directly measured the thick filament compliances of isolated myosin filaments reported in the literature provide a large range of values most likely due to non-consistent experimental conditions or preparation of filaments. Reported estimates from X-ray data from whole muscle are more reasonable, but have an implicit assumption that the compliance is constant along the myosin filament, which is unrealistic given the known variability of myosin filament backbone diameter along its length. By takin in the account the best available structural data regarding the assembly of the thick filament including its variation in diameter from the bare zone toward the tip, number and arrangement of myosin tails and distributed local compliances along the filament more realistic compliance if the thick filaments could be achieved.