Models of thin filament regulation in 3D sarcomere lattice

We have implemented the MacKillop-Gevees 3 state model  (McK-G) and Continuous Flexible Chain Model (CFC) of thin filament regulation into a model of myosin binding in 3D sarcomere lattice.  In Fig. (left) is shown comparison of the force development and relaxation of MG model and flexible chain model. Both models equally well predicted maximum force and Force-pCa relationship (Fig. (middle)), but McK-G model failed to predict relaxation after reduction of      concentration to zero at t=0.5 s (green line at Fig. (left)). In contrast FC model predicted reasonable time course of the relaxation. The reason for the failure of MG model to predict relaxation is size of the cooperative unit. The rigid seven actin monomers TmTn unit has on average 1.8 myosin heads attached at full activated muscle.  Thus if one of the heads detaches, the TmTn cooperative unit remains open until the other attached head detaches too providing sufficient time that some of detached heads can attach to remaining 6 open actin sites. Because the latter process is much faster than detachment of both heads makes the process of closing TmTn units intrinsically slow. Therefore, the TmTn unit needs flexibility to reduce the cooperativity unit size from seven to three actins, as occurs in the CFC model.  On the first site it may appear that the number of crossbridges per TmTn unit is smaller, i.e. about 1.25, what would be the case if all actin sites are reachable, but in real 3D sarcomere geometry some of actin sites are barely accessible due to mismatch of the angular periodicity of actin helical strands and helically arranged myosin crowns. The typical number of these inaccessible sites is 25-30% which increases density of the bound heads per TmTn unit to about 1.8. The failure to relax is also apparent in force development during a series of low frequency electrical stimuli (Fig. (right)). The MG model predicts fast force development even though the stimulus is repeated every 100 ms. In this case myosin binds cooperatively to open units, so effective binding rate dominates the detachment and force development is fast. In contrast CFC model predicts much slower force development, and the ability to reach only a fraction of maximum tetanic force.

Fig. (left):  Force development and subsequent muscle relaxation after reduction of          to zero at  t=0.5 s.  Rigid TmTna7 model (Green line) fails to relax the muscle, while flexible chain predicts reasonable muscle relaxation of muscle after reduction of          to zero (black line).  The prevention of myosin binding at t=0.5 provides the fastest possible relaxation (red line).

Fig. (middle):  Force-pCa curve predicted by 3D stochastic sarcomere model with flexible chain activation model.  KB-pCa relationship obtained from solution (Fig. 18) with some modification is used as activation function in this model.

Fig. (right):  Prediction of force development during sparse activation at 100 ms intervals.  Rigid TmTnA7 unit (MG) model fails to relax between the low frequency stimuli and achieves tetanic contraction, whereas FC model predicts the relaxation between the sparks and achieves only fraction of fully activated tetanic force.

Models of thin filament regulation in solution


Calcium regulation of myosin binding. The calcium dependence of myosin binding to the fully-regulated actin filament can be predicted if the actin affinity of TnI is a known function of the free calcium concentration. The binding of TnI to actin is down-regulated by the binding of two molecules of         to TnC, which has the effect of transferring the corresponding molecule of TnI from actin to TnC. If this process proceeds by an allosteric mechanism, LTI is a decreasing function of          concentration, the decrease being approximately quadratic over a range of concentrations. We have collected transient kinetic data over large range of          concentrations, and these data are reported in our publications.

McKillop-Gevees 3 state model (McK-G).


Molecular structural studies using electron microscopy and 3-D image reconstruction revealed that addition of calcium (        ) causes an 15°-25° azimuthal movement of tropomyosin from the outer to inner domain and that S1 binding causes a further 8°-10° shift. This finding led to notion that the actin filament (F-actin) and Tm-Tn complex can structurally exist in three distinctive states. Subsequently, McKillop and Geeves proposed a three-state model (referred to as the McK-G model) in which several actin monomers (typically 7)  and a Tm-Tn complex form a unit which can exist in three states – blocked, closed and open. In the “blocked” state, myosin S1 binding with actin is prohibited; in the “closed” state, S1 can bind with actin, but cannot be isomerized further to next step; in the “open” state, S1 can bind to F-actin and can be isomerized.  The unit distribution between three states is affected by Calcium (        ) concentration.  Chen, et al. compared predictions of Hill and MG models by fitting the equilibrium binding isotherms and the claimed that these two models are mathematically equivalent. For the Hill model simulations they used a Monte Carlo algorithm which includes nearest neighbor interactions, while for McK-G model simulation they developed a probabilistic algorithm which includes 45 states.

Continous Flexible Chain Model (CFC).


There is now a body of data which shows that the apparent size of the regulatory unit (the number of actin sites simultaneously regulated by tropomyosin) is generally larger than the seven sites associated with one coiled-coil tropomyosin molecule, sometimes by a factor of nearly two. A plausible structural interpretation of this data is that when tropomyosins are strongly coupled end-to-end on actin, the size of the regulatory unit is defined not only by the strength of end-to-end interactions but also by the intrinsic flexibility of the tropomyosin molecule.  From this standpoint, we have proposed a model for regulation of actin-myosin interactions by tropomyosins treated as a continuous flexible chain, loosely confined to one strand of the actin double-helix by electrostatic forces (Fig. 14). An important difference from previous models is that the chain may cover a range of angles on the filament through thermally-driven distortions, rather than being confined to two or three fixed angles as in the steric blocking model of Haselgrove and Huxley or the three-state model of McKillop and Geeves.  Plausible estimates for the bending stiffness of the chain and the strength of its confining potential give regulatory units of 7-12 actin sites as observed and an angular standard deviation of 10-20° which defines the closed state. Myosin binding is inhibited by the majority of configurations of the chain, which is displaced toward the inner domain of actin (upwards in Fig. 14) when myosin binds. The displaced chain then exposes neighbouring actin sites within one persistence length, giving cooperative actin-myosin binding as observed in titrations.

A similar mechanism is proposed for the calcium regulation of myosin binding. Cryo-EM reconstructions of actin-Tm-Tn  filaments in the absence of calcium show that the tropomyosin chain is pinned near the outer domain of actin, presumably by the inhibitory component TnI, suggesting the proposed “blocked state” of the McKillop-Geeves model. Structural models of the troponin complex, e.g. the Herzberg-James model suggest that when one or two calcium ions bind to troponin-C, the distal part of TnI is released from actin and captured by troponin-C. The energetics of the allosteric transitions in TnC which make this possible have been explored in detail by McKay et al. We therefore proposed a model in which the distal part of each TnI is bound either to actin or to the hands of TnC, with a transfer affinity allosterically controlled by the concentration of free calcium.

Mean solution and standard deviation of the Feynman path integral for a given spatial configurations of bound troponin and S1 to actin at low calcium concentrations and short time after mixing regulated actin and S1.

The equilibrium rate constant between blocked and closed state, KB, vs. Ca2+  concentration for both excess actin to S1 and for excess S1 to actin. Both excess actin and excess S1 show sigmoidal KB dependence on pCa with Hills coefficient of 2.0.

The equilibrium statistical mechanics of the fluctuating chain have been explored by path-integral methods and applied to situations where the chain is locally pinned at a positive angle (by bound myosins) or a negative angle (by bound TnI) relative to mean position in the closed state. These predictions were used to define the interaction potential between bound myosins and TnI’s via chain distortions. A second round of statistical mechanics can then predict the equilibrium binding of myosins and TnI’s to regulated actin, and the experimental data can be fitted to the chain model.  We have developed a computational model based on the continuous flexible chain (CFC) theory. The displacement of the chain is determined by the energy of chain kinks, the chain elasticity, and the energy landscape of interaction of TmTn chain with actin. The Feynman path integral was utilized to calculate the mean value of chain angular displacement, f, and the standard deviation around the mean chain position.

Spatial distribution of bound S1 on a single actin strand 700 actin monomers long or 100 TmTn units (horizontal axis). Bound troponin (magenta), free actin site (white), S1 bound to actin in A state (yellow), and S1 in post power stroke state R  (brown). On each panel are shown 40 strands or 20 filaments (vertical axis) for four distinctive times shown in on the right and corresponding normalized fluorescence. Spatially explicit binding scheme provides sizes of cooperative units during S1 binding at low          concentration.

Bound troponin at every seventh actin unit restricts thermal fluctuations to open actin site (closed state at f=0), while unbinding of Tns frees the chain, the fluctuations increase, and therefore effective rate of binding increases too. Our model incorporates structural and kinetic data together thus it can predict unique set of model parameters and fit well with wide scope of experimental data. Our results showed that only one parameter of our CFC model is needed and the CFC model can uniquely predict these parameters and excellently fit the experimental data. Furthermore, spatially explicit positions of actin sites, with bound troponin and bound S1 in either A or R state provide explicit information on the size of cooperative unit during binding process.  The evolution of myosin binding to actin is shown here.