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Chapter 7:

Modelling Simple Structures: Actin

E lineis fit robur
(Strength comes from cords.)
_____________- anon


Actin is a common cytoskeletal protein that has been studied in great detail (see Chapter 2). It forms long, filamentous strands that are extremely common in all eukaryotic cells. It is particularly common in muscle tissue where, together with the protein myosin, it provides the mechanical forces required for muscle action (1).

This chapter looks at how the nanosim program can be used to simulate the self-assembly of actin. It examines the existing theory and experimental evidence, and shows how this can be processed to produce a behavioural model for a single actin monomer. It then describes the use of the simulator to model actin self-assembly, and reviews the program's results, comparing them with other experimental evidence.

The Different States of Actin

Individual actin proteins in the unbound state are known as g-actin ("globular" actin), while actin bound within an actin strand is known as f-actin ("filamentous" actin). Actin can self-assemble in vitro, and has the property that both assembly and disassembly occur at a faster rate at one end of a filament, called the "plus" end, than at the other end of a filament, the "minus" end (Fig. 7.1) (2).

This difference in assembly and disassembly rates at either end can lead, at appropriate concentrations, to a process known as treadmilling, where a particular filament may be growing at one end while shrinking at the other.

A further complication is that, in common with many self-assembling proteins, actin dephosphorylates or hydrolyses when it binds. Phosphorylated actin has a bound adenosine 5' tri-phosphate molecule (ATP), a common energy-providing molecule used in a large number of cellular processes. This ATP molecule makes the actin protein far more likely to bind to other actin proteins. Some time after being bound, in a process that is not yet fully understood, the bound ATP molecule hydrolyses, losing a phosphate ion and becoming the less energetic adenosine 5' di-phosphate (ADP). This in turn causes the actin protein to become less securely bound.

It has been established that actin filaments usually terminate with phosphorylated ATP-actin, but whether the individual molecule dephosphorylates over time, or upon the addition of another actin molecule, is not yet completely established (a similar process, and a similar debate, occur as regards the tubulin protein examined in Chapter 8).

The Structure of Actin Filaments

The structure of actin has been extensively studied using electron microscopy and image analysis and, despite many experimental difficulties, the nature of actin filaments is now well understood (3). It can be considered as a long two-stranded helix with a repeating length of approximately 36- 40 nm (the length 'L' in Fig. 7.2 (4)).

Actin Polymerisation and the Oosawa Model

The assembly of actin is one of the finest examples of Oosawa's model of protein polymerisation in action, closely fitting the mathematical predictions made by the model (5) (cf Chapter 5). Oosawa's model predicts that the form of the polymerisation curve (a graph of the amount of the actin bound in filaments, as opposed to unbound, over time) will be the same for different environmental conditions, differing only in the rate that the reaction proceeds. Measurement of polymerisation is usually done via the viscosity of the solution, and results for the polymerisation of actin can be shown to have the same shape when plotted against logarithmic time, being simply offset by different amounts depending on whether the environmental conditions favoured rapid or slow polymerisation (Fig. 7.3) (6) :

The Oosawa model predicts that the slope of these curves at intermediate values will be relatively constant, approaching the on-rate times the concentration minus the off rate; konc - koff. (This characteristic slope is more evident on a linear scale than on the logarithmic scale above.)

Another prediction of Oosawa's theory is that the length distribution of actin filaments is initially a Poisson distribution at the start of polymerisation, moving to an exponential distribution when a solution has reached steady state. This is confirmed by painstaking experimental studies that have catalogued the length of actin filaments, and have found the expected exponential distribution (Fig.7.4) (7), (8):

Suitability of Actin for Simulation

These theoretical details, in conjunction with the wealth of experimental data, make actin an excellent candidate for trialing the nanosimulator, to see whether it reproduces correctly the behaviour of this relatively simple self-assembling protein. Actin has been previously simulated mathematically with good results, and a number of sophisticated models (including those modelling actin-associated proteins) have been developed (9), (10), (11). As far as the author is aware however the computational technique of simulating large numbers of individual actin monomers as described below is new.

Setting up the Simulation

Experimental studies of actin have provided accurate details of the assembly and disassembly rates of the growing filament ends. To test the simulation, an arbitrary in vitro environment is chosen for which good experimental evidence is available. The current simulation uses data from Pollard (12), who used a standard experimental environment conducive to filament growth, at 37C, and obtained the following figures (also used in figure 7.1):

  ATP-Actin ADP-Actin
+ end - end + end - end
kon (M-1s-1) 11.6 1.3 3.8 0.16
koff (s-1) 1.4 0.8 7.2 0.27

Table 7.1 Actin association and disassociation rates (13)

Some interesting features of these figures deserve note. First, although ATP-actin is the most active and inclined to bind, it is obviously possible at higher concentrations (i.e. around the 2 M level and above) for ADP-actin to polymerise into filaments. Second, although ATP-actin is generally more likely to bind and less likely to break away than ADP-actin, in a curious inversion ADP-actin is less likely to break away from the slow-growth, "-" end of the filament.

Using these figures, a .pddf file (the "protein dynamic description file" used by the nanoscale simulation program) was constructed, that uses a four-state model for actin (Fig. 7.5):