Project 2: muscle activation



Activation is critical to muscle function.  For example, in order for a heart to beat, a particular cell must actively generate force and/or motion at some times and be passive at other times.  This activity is regulated by calcium.  When a nerve signal reaches a muscle cell, calcium is released into the cell.  This calcium is bound by troponin, a globular protein associated with actin.  Troponin, in turn, affects the position of tropomyosin, a filamentous protein that wraps around actin.  It is the azimuthal position of tropomyosin on actin that regulates myosin binding to actin (see Fig. 1A).


Tropomyosin exists in a dynamic equilibrium between three states on actin: blocked, closed and open.  In the blocked state, tropomyosin occupies a position on actin where myosin binding is sterically inhibited.  Calcium binding by troponin favors a shift in tropomyosin's position to the closed state, where myosin binding is more favorable.  Tropomyosin is further shifted into the open state when a myosin molecule binds to actin.  The shift in tropomyosin position due to myosin binding facilitates the binding of other myosin molecules.  Importantly, this effect is local; the binding of one myosin only accelerates the binding of nearby myosin (see Fig. 1B and C).





Figure 1: Cartoons of regulation.  A. Calcium binding by troponin (black blobs) displaces tropomyosin from the blocked (black) to the closed (gray) position.  Myosin binding further displaces tropomyosin into the open position (white).  B.  Myosin binding to actin causes a local displacement in tropomyosin.  C. In an ensemble, myosin's displacement of tropomyosin introduces local coupling between molecules.  For example, myosin molecule 3 would bind to actin more readily than myosin molecule 5.







These local interactions violate a fundamental assumption of molecular muscle models: that each myosin molecule acts independently of its neighbors.  How can we model ensembles of locally coupled myosin motors?  The goal of project 2 is to develop accurate and efficient methods to model these systems.


Below is a summary of our current progress toward this goal.  Details can be found in the papers (referenced below).


Publication 1: simplified conditions


Reference: Walcott, S., A differential equation model for tropomyosin-induced myosin cooperativity describes myosin-myosin interactions at low calcium. Cellular and Molecular Bioengineering, Volume 6, pages 13-25. 2013. PDF


Developing a model of locally coupled myosin ensembles is challenging.  As a starting point, we developed a model that applies under simplified, but experimentally relevant, conditions.  Specifically, we consider low ATP concentration (which eliminates force-dependent detachment) and low or high calcium.


Figure 2: Modeling experiments with locally coupled myosin ensembles.  A. A schematic of the motility assay.  Myosin is affixed to a 2D surface.  Fluorescently labeled regulated thin filaments are added.  In the presence of ATP, the filaments move and their average speed can be measured.  B.  Measurement of filament speed as a function of ATP at high and low calcium.  Data were fit with the model, and the model was validated with comparison to computer simulations.




Under these conditions, Kad et al. (2005) observed a striking result in the in vitro motility assay.  In this experiment, myosin is adhered to a surface and regulated thin filaments are subsequently added.  In the presence of ATP, these filaments move and their average speed can be measured (see Fig. 2A).  When calcium concentration is high (pCa 4), sliding speed increases roughly linearly with ATP.  This result is expected since, under these low ATP conditions, ATP binding limits myosin's detachment from actin.  However, when calcium concentration is low (pCa 8), a biphasic response is observed.  Speed initially increases linearly with ATP, but then abruptly starts to decrease with increasing ATP (see Fig. 2B).  Local coupling between myosin molecules underlies this biphasic response.


In this paper, we developed a simple way to describe local coupling with only two parameters.  We then proposed two methods to model locally coupled myosin ensembles: the weakly-correlated and the linear theory. The linear theory is consistent with detailed computer simulations and fits motility experiments at high and low calcium (Fig. 2B).  Unlike computer simulations, the theory is sufficiently accurate and efficient to allow both parameter estimation and a sensitivity analysis.

Publication 2: a complete model


Reference:  Walcott, S., Muscle activation described with a differential equation model for large ensembles of locally coupled molecular motors. Physical Review E, Volume 90(4), pages 42717. 2014.  PDF


In this paper, we developed a theory for locally coupled myosin ensembles that applies to physiologically relevant conditions.  The basic idea of the theory is to divide the thin filament into two phases.  In the "active" phase, myosin molecules are uncoupled and bind to actin without any interference from the regulatory proteins.  In the "inactive" phase, myosin molecules are not bound to the thin filament.  It is only at the border between the two phases that local coupling between myosin molecules is important.


The size of the active and inactive phases is described by a set of linear ordinary differential equations (ODEs).  The uncoupled myosin molecules in the active phase are described by the integro-partial differential equations (iPDEs) we derived in Project 1.  We can then describe the entire system with a set of coupled ODE-iPDEs.


The derivation of these equations requires the assumption that the thin filament can be divided into two phases.  We tested the validity of this assumption by performing simulated experiments.  In one of these simulated experiments (Fig. 3), we applied a time-varying force to a thin filament.  Using Monte-Carlo methods, we explicitly simulated the behavior of 1000 myosin molecules as they interacted with this thin filament.  The position of the thin filament and the proportion of myosin molecules bound to the thin filament showed complex behavior as a function of time.  The coupled ODE-iPDEs of our theory reasonably reproduced this complex behavior (Fig. 3).  We therefore conclude that our assumption is reasonable.






Figure 3: The theory is consistent with simulations, suggesting that our assumptions are reasonable.  We simulated the behavior of a large ensemble (1000 molecules) of myosin interacting with a thin filament.  In these simulations, a time-variable force was applied to the thin filament (bottom).  The theory successfully predicts the proportion of myosin molecules bound to the filament (middle) and the position of the thin filament (top).







A desire to model calcium-dependent muscle activation was our primary motivation for developing this theory.  So, can we use this theory to describe activation experiments?  In publication 1, we showed that it could fit measurements of regulated thin filament speed as a function of ATP at low and high calcium in the motility assay.  This new theory allows us to describe experiments at variable calcium.  For example, we used the model to fit measurements of isometric force as a function of pCa (the negative log of the calcium concentration) (Fig. 4A).  We then predicted the results of experiments measuring thin filament speed as a function of pCa.  The model is consistent with recently published measurements (Fig. 4B).


Our theory efficiently describes the behavior of locally coupled myosin ensembles (Fig. 3).  The theory reasonably fits experimental measurements at the fiber scale, and can be easily used to estimate parameters (Fig. 4A).  Finally, the model successfully predicts experimental results (Fig. 4B).  The success of the model raises the hope of developing a micro- to macro-scale muscle model.



Figure 4: The model describes experiments at variable calcium.  A. Isometric force as a function of pCa (negative log of the calcium concentration) for a muscle fiber.  Data are reasonably fit by the theory.  The theory has a single parameter.  B. Prediction of motility speed as a function of pCa is consistent with experimental measurements.


Primary Collaborators


Ned Debold

Neil Kad