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 |