Skeletal muscle actomyosin

Almost all processes in a living cell are performed by molecular machines. Most of these are made of proteins (with the notable exception of the ribosome, which is mostly made of RNA). One class of molecular machines that is particularly well suited to a physical description is molecular motors – those machines that generate movement. This lecture will show how a molecular motor can be modelled using the formalism of diffusion and chemical reactions (Slide 1). This is essential because, unlike macroscopic machines, molecular machines operate at energy scales close to the thermal energy, kBT, and are therefore subject to continuous thermal fluctuations. These fluctuations are an essential ingredient in how the motors work. Actomyosin, the molecular motor that makes skeletal muscles move, will be used as an example.

Non-equilibrium thermodynamics considers forces and their conjugate fluxes (Slide 2). Thermodynamic forces drive a system to change, and that change is measured as the conjugate flux. The product of a force and its conjugate flux is power. Coupling between non-conjugate forces and fluxes is called free energy transduction. Motors couple non-mechanical forces to mechanical fluxes; generators, such as ATP synthase, are motors running in reverse.

Muscle myosin (Slide 3) is a molecular motor that causes muscle contraction by moving relative to its ‘track’, a filament of actin. It transduces energy from the hydrolysis of ATP, a chemical fuel, to perform mechanical work. Myosin, with F1-ATPase and kinesin, is one of the best-understood molecular motors. The diagram in Slide 3 reproduces the overall scheme of biological free energy from ‘Biological Energy’ Lecture 1 (pmf is protonmotive force).

Muscle myosin is a dimer with two identical motor heads that act independently. Each myosin head has a catalytic core (a domain in the middle of the part of the molecule – blue in the movie – that binds actin) incorporating a nucleotide binding pocket (a cavity in the core that binds ATP and couples its hydrolysis to shape changes of the whole molecule), attached to a lever arm (yellow in the movie). A coiled-coil rod (grey in the movie) ties the heads together and links them to the thick filament (top in the movie), which connects many myosin molecules in parallel. The motors pull on the thin filament (bottom in the movie), a helical polymer of actin.

The movie illustrates the mechanism of skeletal muscle actomyosin, as described below:

  1. The nucleotide binding pocket of a detached myosin head contains ADP and inorganic phosphate (Pi). In this state it has a weak affinity for actin.
  2. Once one head docks, Pi is released...
  3. ... which strengthens binding and triggers a force-generating power stroke that swings the end of the lever arm through about 5 nm, moving the actin filament relative to the thick filament.
  4. After the power stroke, ADP dissociates and ATP binds to the empty nucleotide binding pocket, causing the head to detach from the actin filament.
  5. ATP is hydrolyzed in the detached head …
  6. ... which resets the lever arm back to its pre-stroke state. The cycle can then repeat. The actin filament does not slide back when released because many other myosin molecules are still attached, holding it under tension.

The rest of this lecture will explain in more detail the physics of how this molecular motor works and how it can be quantified.

1.1 The mechanism of muscle contraction

Muscle is highly organized, on length scales from molecular to macroscopic. Many molecules are connected in parallel and in series (see Slide 4), so that individual molecules capable of exerting piconewton forces and with working strokes of 5 nm produce the macroscopic forces and displacements of whole muscles.

Muscles have a repeating unit called the sarcomere, approximately 2 µm in length. Arrays of thin (actin) filaments, anchored together at the Z discs, slide past interdigitated thick (myosin) filaments. Myosin heads make crossbridges between the thick and thin filaments – their cycle of binding, pulling and release drives the relative motion of the filaments, which causes the sarcomere to contract. A F Huxley’s discovery of this mechanism led to his award of a Nobel Prize for Medicine in 1963 (shared with J C Eccles and A F Hodgkin, who also made discoveries concerning mechanisms in nerve cells).

The following movie shows a sarcomere contracting. Multiple sarcomeres in parallel allow the exertion of a large force, while fast contraction is achieved by having multiple sarcomeres in series. Titin is a passive spring (see ‘Biological molecules’ Lecture 1) that stabilizes the sarcomere and acts as a shock absorber.

In the in vitro gliding assay shown as a still in Slide 5 and a movie below this, actin filaments glide on a carpet of myosin motors immobilized on a glass slide. Actin filaments are visible because they are fluorescently labelled. (The actin and myosin shown here are actually from different species, so it is not a natural juxtaposition, though it does still work.)

The gliding direction is determined by the polarity of the actin filaments. Unlike ds-DNA backbones, actin filaments are asymmetric under reversal of their ends. The polarity of an actin filament is defined by binding fragments of myosin to them and looking at the ‘decorated’ filaments with an electron microscope. Based on the appearance of these decorated filaments, one end is called ‘barbed’ (+) and the other ‘pointed’ (-). In the in vitro assay, myosin heads are randomly oriented on the surface. Only myosin molecules that are oriented correctly can bind an actin filament, whereupon the power stroke tries to move them towards the barbed (+) end of the filament. Because the myosin molecules are stuck to the surface, the end result is that the actin filament moves with its pointed (–) end forwards.

Each actin filament binds many myosin molecules at a time. Each myosin molecule only binds actin intermittently, but because there are many myosin molecules the actin stays bound to the surface. To see a single actin–myosin interaction in vitro (in glass, as opposed to in vivo, in life), the actin and myosin need to be held close together (as in muscle). Single-molecule measurements with optical tweezers have achieved this and will be described in Lecture 3.

1.1.1 The mechanochemical cycle of actomyosin

To make a mathematical model of myosin, we first write down what we understand about its operating cycle as a series of more or less well-defined states and the transitions between them. This ‘cartoon’ of the crossbridge cycle (Slide 6) is the combined output of decades of biological and biochemical research.

The stages depicted are:

6: Without bound ATP, myosin binds actin tightly. This state is called ‘rigor’: when ATP is removed, muscles are stiff with all myosin crossbridges tightly bound.

6→1: When ATP binds, myosin loses its affinity for actin, rapidly leading to…

1→2: unbinding, which in turn allows…

2→3: hydrolysis of bound ATP coupled to the ‘recovery stroke’, where the neck-linker domain tilts about 20° relative to the bound head. After the recovery stroke ...

3→4: myosin can rebind the first available actin molecule, triggering ...

4→5: phosphate release, which triggers the power stroke. This is mechanically the reverse of the recovery stroke, except that myosin is bound to actin and thus a force can be generated between them.

5→6: After the power stroke, ADP release completes the cycle, leaving the myosin molecule ready for the next cycle.

ADP is adenosine diphosphate. See Lectures 1 and 2 in the ‘Biological energy’ topic.)

Overall, one cycle as shown catalyses the hydrolysis of one molecule of ATP and couples this to one power stroke. The power stroke pulls the myosin tail (black line) 5 nm relative to the actin filament. The combined action of many myosin heads, each either pulling or, unbound, waiting to pull, causes relative sliding of the myosin and actin filaments, shortening the muscle. All transitions are reversible but the free energy of ATP hydrolysis leads to net flux in the direction shown by the arrows.

The state of a myosin molecule can be described by the contents of the nucleotide-binding site (ATP, ADP+Pi, ADP, empty) and the relative positions of the actin and myosin filaments, determined by whether the head is bound and its shape. Transitions between states, which differ both in their chemistry and in the mechanical output variable, are the essential core of the mechanism of any molecular motor. Transitions between two states that depend simultaneously on two different types of free energy are the basis of any free energy coupling mechanism.

Slide 7 shows X-ray crystal structures of states of the myosin model. Each myosin molecule contains of the order of 103 residues and 104 atoms – to define the state of the molecule completely would require the specification of, at least, the dihedral angles (ψ, φ) for each residue, the contents of the nucleotide binding pocket (and their coordinates) and a detailed description of the myosin–actin interface, if bound. The only model with this level of detail is a molecular dynamics simulation, in which the interactions between atoms are described by empirical force fields and the equations of motion for each atom are integrated numerically. Full atomic simulations are very demanding – trajectories of small proteins can typically be calculated for nanoseconds at most, whereas cycle times are of the order of milliseconds. To make these models more tractable, atomistic detail must be replaced by a lower-resolution description of the motor – in molecular dynamics this process is referred to as ‘coarse graining’.

Each ‘state’ of the motor is an abstraction that represents a huge ensemble of related conformations (microstates) with similar global chemical and mechanical properties – for example, having ATP bound and having the lever arm at a particular angle.

1.2 Mechanisms and models of molecular motors

1.2.1 Free-energy landscapes for molecular motors

In Slide 8 we have reduced the complexity of actomyosin to two abstract variables. The mechanical coordinate measures the distance along the muscle between any arbitrary markers on the actin and myosin filaments; the reaction coordinate describes progress in the overall reaction of ATP hydrolysis. In this particular slide the actomyosin power stroke is shown. State 5 has an equilibrium position shifted (5 nm) compared with state 4, due to motion of the myosin lever arm. Elasticity in the system means that deviations from the equilibrium position cause increases in free energy, which are locally quadratic. The lever arm motion is triggered by phosphate release, one of the reactions that make up ATP hydrolysis.

In general, action of the motor could be modelled as diffusion in the two-dimensional space of Slide 8. However, the space is not isotropic: the diffusion constant along the reaction co-ordinate is much higher than that along the mechanical coordinate. This is because reactions involve motion of small molecules such as ATP and Pi, or a few side-chains around the ATP-binding pocket and the lever-arm hinge. By contrast, the mechanical coordinate involves motion of the entire muscle. Because of this, it is sensible to model chemical changes along the reaction coordinate as instantaneous jumps between discrete states (as in Eyring theory, see the supplementary lecture ‘Chemical reaction kinetics and equilibrium’) while retaining a continuous treatment of the mechanical coordinate.

The two horizontal sections through the two-dimensional energy landscape in Slide 9 represent the free energy of the system as a function of the mechanical coordinate x in states 4 and 5 – that is, before and after phosphate release.

The transition is represented by a jump between these free energy surfaces, during which the mechanical coordinate does not change. The subsequent trajectory of the system is stochastic (it depends on thermal fluctuations) but on average it will relax towards the energy minimum of state 5 and the free energy released is available to do work to drive this mechanical step. Like any other chemical reaction, the transition between states is a Poisson process: the transition rate corresponds to the probability per unit time that a transition will occur.

1.2.2 A closer look at the reaction coordinate

This section is a review of chemical transition theory in which the basic ideas discussed in the lecture ‘Chemical reaction kinetics and equilibrium’ are applied to actomyosin.

Slide 10 shows a simpler free-energy surface than that in slides 8 and 9, with two well-defined minima at the same position coordinate. This simple description is justified empirically and by Eyring/Transition State Theory. We have introduced this as an intermediate step to make the later analysis easier to follow.

We may collect the ensembles of microstates near the minima into effective states X, Y and plot a free-energy profile for the transition between them. The Arrhenius equation gives an approximate description of the temperature dependence of the transition rate, which depends exponentially on the height of the energy barrier G*:

Equation 1
Equation 1

Equation 2
Equation 2

The pre-exponential factor, A, is weakly dependent on temperature and may be interpreted as an attempt frequency.

For every trajectory in configuration space there is a time-reversed trajectory. At equilibrium the principle of detailed balance applies: there is no net flux of probability along any trajectory, which means that the probability that the system will evolve along a given trajectory is exactly equal to the probability that it will return along the same trajectory. For the transition between states X and Y, this simply means that, at equilibrium, there is no net flux over the potential barrier. This is satisfied by the first-order rate equations used to describe the X↔Y system because the ratio of rate constants is inversely related to the ratio of equilibrium occupation probabilities:

Equation 3
Equation 3

(See the supplementary lecture ‘Chemical reaction kinetics and equilibrium’.)

1.2.3 Position-gated transitions

Another essential ingredient of an efficient mechanochemical cycle is that chemical reactions should occur mostly at positions where they will lead to work and motion of the motor.

Slide 11 explains a key feature of how molecular machines work. By discretising the chemical coordinate we have lost information about the shape of the free energy surface between states that governs the probability of a chemical transition. The three vertical sections of the energy landscape (a, b, c in Slide 11) represent the free energy of the system as a function of the chemical coordinate (reaction coordinate) for different, fixed values of the mechanical coordinate. The transition rate is position-dependent: the barrier is significantly higher at positions a and c than at b. The narrow saddle in the free energy surface between states 4 and 5, which is close to the minimum energy of state 4, is a gap or ‘gate’ in the free-energy barrier between the chemical states, which determines the motor position (mechanical coordinate) at which the probability of phosphate release is highest. Physically, the gate is probably something straightforward –like an alignment of two components that have to be close for the reaction to occur.

1.2.4 Introducing a quantitative model of actomyosin

The reaction rate constants and free energies shown in Slide 12 are assembled from a great deal of experimental data plus a certain amount of educated guesswork (see Howard Chapter 14 for details). These rates are averages, measured under a variety of physiologically relevant conditions. The experiments do not control the mechanical coordinate, and best represent the equilibrium position (minimum) of each state.

States in the mechanochemical cycle are defined by whether actin is bound to myosin, and by which if any of ATP and its hydrolysis products are bound, which in turn determines the most stable shape of the myosin molecule. States are numbered 1–6 for future reference. Typical rate constants for transitions between the states are given, as well as free energies relative to the reference state labelled 2.

Note that binding is the slowest forward step (highlighted by the red circle): each head spends most of its time unbound – idling, while other heads pull. This motor is described as having a ‘low duty ratio’. This model will be developed and discussed in further detail in Lecture 2 of this topic.