Questions

1 (a) The mechano-chemical cycle of the actin-myosin molecular motor in skeletal muscle is shown in Figure 1 (the same diagram appears in Slide 6 of Lecture 1). Describe the states and transitions in the cycle, and explain how they couple the hydrolysis of ATP to active contraction of muscles.

Figure 1


Figure 1 Acto-myosin crossbridge cycle

(b) Experimentally measured (or estimated) average rate constants and relative free energies for the cycle arc given in Figure 2 (the top-left state in Figure 1 corresponds to AMT in Figure 2. A = actin, M = myosin, T=ATP, D=ADP, P = phosphate).

Figure 2


The cycle can be modelled as a sequence of instantaneous chemical transitions between the states. Make a labelled sketch of the free energy profiles for each state in the cycle, as functions of the relative position of actin and myosin filaments. Indicate with arrows the transitions between states. Using the same position axis, sketch also a graph of the rate constants for these transitions.

(c) The model can be reduced to a minimal form that includes one binding powerstroke with rate constant kon (combining MDP →AMDP → AMD) and one release step with rate constant koff (combining AMD → AM → AMT → MT → MDP). Make similar sketches to those in b) above for the reduced model.

(d) Use the minimal model to show that the average force per crossbridge is given by

Equation


where d is the distance between accessible binding sites for a myosin head. Assume that the muscle fibre slides at a constant velocity v due to the combined action of millions of crossbridges, that the crossbridge can be described by a linear spring of stiffness κ in all bound states, that binding occurs only in a very small range of positions between x0 and x0 + Δx (x0 + Δx < 0) where x = 0 is defined as the position where the post-powerstroke crossbridge is relaxed, and that release is possible only when x > 0.

2 Go to www.pdb.org (the Protein Data Bank, PDB) and explore links under ‘PDB-101’ at bottom left, including ‘Understanding PDB Data’ and ‘Molecule of the Month’ (MOM).

To go straight to a database entry for a particular structure click on its PDB ID in MOM or enter the ID in the search box near the top of the page. Use an interactive viewer, e.g. Jmol, to look at structures – see the options given under the thumbnail picture of the molecule on the RHS of the structure page. You may have to install Java on your computer. N.B. Alternative viewers (available on the web, and see menu) include Rasmol, the Swiss-PDB Viewer and Pymol– these make it easier to change the way the data is displayed (Pymol was used to make the images used in the lectures). Suggestions for display settings below refer to Jmol.

Read the entries for the following molecular machines in the MOM archive.

(a) Kinesin (MOM April 2005; PDB ID 3kin)

What molecule is bound to each of the two heads?

Measure the distance between them (right click in Jmol for an appropriate menu). This distance is approximately the step size, and kinesin can walk at 100 Hz (on average).

How long does it take to transport a vesicle of neurotransmitter from a cell body at the base of your spine to your big toe? Roughly how long would it take, on average, for a typical globular protein with diffusion constant D = 10–10 m2 s–1 to diffuse the same distance?

(b) Actin and myosin (MOM June, July 2001)

The force-velocity relationship for muscle derived in Lectures 1 and 2 depends on seven structural and kinetic parameters. Discuss the number of parameters that are required to describe the instantaneous configuration of a single myosin head.

3 Describe two single molecule experimental methods that have revealed features of the mechanism of biological molecular machines, and outline at least one discovery made with the aid of each method.

Answers

This problem covers the use of “Brownian dynamics” to model molecular motors. Brownian dynamics, which consists of defining a small number of mechanical co-ordinates and reducing all other degrees of freedom of the system to instantaneous transitions between a set of discrete states, is a substantial part of the course. The resulting models are described by reaction-diffusion equations - partial differential equations that cannot usually be solved analytically. This problem uses the simplifying assumptions of constant velocity and steady-state probabilities – realistic in skeletal muscle – to reduce the problem to a simple ordinary differential equation. in one spatial variable that can be solved relatively easily. The result compares well with measured force-velocity curves in muscle. This material is covered in Howard Chapters 14 and 16, and in the lectures.

1 (a) Overall: one cycle as shown catalyses the hydrolysis of one molecule of ATP and couples this to one powerstroke. The powerstroke 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.

Step by step:

6 → 1: without bound ATP, myosin binds actin tightly. This state (6) is called “rigor”, as it leads to stiff muscles with all myosin cross-bridges tightly bound when ATP is removed- as in rigor mortis. When ATP binds, myosin loses its affinity for actin, rapidly leading to

1 → 2: un-binding, 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 re-bind the first available actin molecule, triggering...

4 → 5: phosphate release, which triggers the powerstroke. 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 powerstroke ADP release completes the cycle, leaving the myosin molecule ready for the next cycle.

See Lecture 1 ‘The Molecular Mechanism of Muscle’, Section 1.1.

(b)

Figure 3 Free-energy profiles of states in the acto-myosin cycle
Figure 3 Free-energy profiles of states in the acto-myosin cycle


The energies of the unbound states and the minima of the bound states correspond to the energies in Figure 2: 0, –12, –15, –17 kBT, and the overall ATP hydrolysis energy of the cycle –25 kBT.

The numbers in the table come from a range of measurements that lack position information, which best represent the equilibrium position (minimum) of each state. It would probably be more accurate to draw the rate-constants as curves rather than step-functions, but the key idea is that binding and unbinding are only possible at certain positions that favour completion of forwards powerstrokes. The formalism of this diagram, called Brownian Dynamics, explains how the cycle shown in the cartoon arises naturally from the energetics and kinetics (ie. rate constants and their dependencies) of the system.

“Binding” refers to the rate constant k34. “Un-binding” could be prevented before the end of the powerstroke by a rate constant of the form shown (i.e. very slow for x < 0) for at least one of the sequential transitions i>k56, (ADP release), k61 (ATP binding) or k12 (actual un-binding). (See Howard table 14.2 and surrounding pages for a discussion of which on one it is likely to be.)

The stiffnesses of the bound states are not well known, but should be sketched such that the energy profiles of states 5, 6 and 1 allow the power-stroke to proceed “downhill”. If the muscle is exerting a force, then the work done corresponds to a tilting of all energy profiles. This allows a rough estimate of the maximum force each head can exert…

~15 kBT / 5 nm = 15 ×4 pN nm / 5 nm = 12 pN.

It might be useful to discuss why free energy is “wasted” in the un-binding step 1 → 2. This avoids dragging crossbridges: as soon as the powerstroke is over, rapid and irreversible release is desirable. Thus muscles generate heat, and the cycle is intrinsically inefficient. Other similar molecular motors (including some non-muscle myosins) can be much more energy efficient – skeletal muscle acto-myosin is optimized to work fast, not efficiently.

Figure 4 A minimal model for muscle acto-myosin
Figure 4 A minimal model for muscle acto-myosin


See Lecture 2 ‘Modelling a molecular machine’ Section 2.2 and Slide 5.

The minimal model groups together sequential states where there is no mechanical change into 2 “compound” states: “bound” and “unbound”. There is absolutely no problem with this, remember that each state in the first place was defined pragmatically as an ensemble of conformations of the whole system which shared a certain property of interest. Thus the compound states are every bit as valid as the original ones, the difference being that their definition chooses to ignore the details of the chemistry (ATP hydrolysis) that provides the free-energy changes necessary to drive the cycle forwards.

(d) For a filament moving at constant speed, the reaction-diffusion equation for Pbound(x, t), the probability density that a given head is bound (see lecture notes) satisfies:

Equation


for steady state. (See Lecture 2 ‘Modelling a molecular machine’ Section 2.2.)

(i) As the head traverses the binding zone (x0< x < x0+ Δx):

kon = const

koff = 0

initial condition P = 0

Equation


so the probability of binding before leaving the target zone is:

Equation


(Pb is unchanged in the range x0<x< 0.)

(ii) Unbinding (x > 0):

kon = 0

koff = const

initial condition P = Pb

Equation


Figure 5
Figure 5


Force exerted by crossbridge is –κx, i.e. forwards for x0 < x < 0 and backwards for x > 0. Work done, on average, per binding site traversed is given by:

Equation



NB The second integral involves integration by parts:

Equation


The average force, F, per binding site is

Equation


Figure 6 (below) and Slide 8 in Lecture 2 illustrate the agreement between theory and experiment.

Figure 6
Figure 6


2 (a) Kinesin (MOM April 2005; PDB ID 3kin)

Each head binds adenosine diphosphate (ADP) (zoom in and count the phosphorus atoms), i.e. spent (hydrolysed) fuel.

Distance between ADPs is approximately 9 nm, so walking speed ~ 1 μm s–1 and time taken to get to big toe is ~10 days.

Diffusion: dimensionality is not well defined but it is reasonable to use

〈x2〉≈Dt

or any version of

〈x2〉 = 2dDt

where d is dimensionality.

Time for a single protein to diffuse 1m is of the order of 1010 s ≈ 300 years. (1 year ≈ π x 107 s).

For molecular motors, see the book by Howard, and Berg et al Chapter 34.

(b) The parameters of the model are those that appear in the formula for the average force in Equation 10 (Slide 7) of Lecture 2 ‘Modelling a Molecular Machine’:

kinetic parameters kon, koff , v

structural parameters x0, Dx, κ, d.

The kinetic parameters relate to transformations, so are not linked to “the instantaneous configuration of a single myosin head”.

x0 and Dx are properties of the actin filament, specifically related to the locations where myosin can bind. And d is the periodicity of binding sites on actin. Strictly speaking only κ, the stiffness of the crossbridge, is a property of the instantaneous configuration of a single myosin head. However note that the head can take two distinct configurations, pre- and post-powerstroke. The angle at the neck changes between these, changing the equilibrium position of the bound state by the powerstroke distance x0. Remember that x0 is the distance the thick filament travels relative to actin between myosin binding pre-powerstroke and the end of the powerstroke. Thus it is set by the powerstroke, which is a property not of the instantaneous configuration of a single myosin head, but of the difference between two separate instantaneous configurations.

3 This question asks for an essay on the material presented in Lecture 3 ‘Single Molecule Methods’, plus any further information from wider reading