Visual representation

Molecule drawing algorithm

In many cases, the three-dimensional structure of a protein molecule contains several distinguishable domains, which may again contain subdomains. The shape, the size and the location of these domains and subdomains are determined by the way that the protein molecule is folded. The structure of a protein molecule could be described in terms of domains even when the structure data is only at low resolution. In this study, the three-dimensional shape of a protein molecule is represented by an isosurface of the electron density of the molecular system. Domains are the basic graphic objects involved in the generation of the surface.

For a supermacromolecular system such as the actin filament, suppose that there are N protein molecules bound to each other and that the molecule contains domains (). The position of molecule can be described by a three-dimensional vector . The position of the domains in molecule with respect to can be described by a set of vectors (). In a given orthogonal coordinate system with a unit vector set {}, the position vector of domain in molecule can be expressed as:

We assume that the electron distribution of each domain can be approximated by a three dimensional Gaussian function and that the electron density function of the whole system equals to the summation of the contributions from every individual domains. In other words, the electron density of the supermacromolecular system can be expressed as a function of the space vector ():

where is a normalization constant and the parameters , and are the dimensional measurements of domain in molecule along the directions , and , respectively.

The isosurface of the electron density of the molecular system is drawn according to the following procedure in general: (1) calculate vectors and based on the current structural understandings of the molecular system; (2) specify parameters , and based on the measurements of the domains and the bound connections between them; (3) if possible, calculate the alterations on , , , and corresponding to the distortion on the shape of domains during the configurational change of each molecule; (4) determine the threshold value of the electron density for an isosurface of Eq. (4); (5) calculate the vertices of and the surface normals of the isosurface according to an appropriate algorithm such as the ``marching cubes''; (6) draw the isosurface by using vertex subroutines in SGI Graphics Library; (7) if possible, compare the calculated image with the 3-dimensional reconstruction data of electron microscopy; (8) repeat steps (2) to (7) until a satisfactory representation is reached.

F-actin helix and myosin subfragment-1

The F-actin helix can be regarded as a two-stranded structure with a half-pitch of about 37 nm and a diameter of 9 to 10 nm. Electron microscopy and three-dimensional reconstruction techniques have been used to visualize the geometric shape of real F-actin filaments. The bulk of the actin monomer is about 5.5 5.5 3.5 and is composed of two domains. The `large' inner domain contains subdomains 3 and 4, whereas the `small' outer domain contains subdomains 1 and 2. The long axis of the monomer lies roughly perpendicular to the filament axis. The myosin subfragment 1 binding sites lies on subdomain-1 of actin. The myosin head approaches the actin filament tangentially and binds to a single actin.

The graphic representation of F-actin is constructed based on a reduced atomic model. The electron density of each of the four subdomains in an actin monomer is approximated by a 3-dimensional Gaussian function. The parameters , and are determined based on the mass of the subdomains and the putative bonds between them. The density function of the F-actin at a given point is then calculated according to Eq. (4). We assume that the contribution to the density function from actin mononers that lie beyond the five nearest ones are negligible. Therefore, in the calculation of Eq. (4), we only consider the 20 subdomains in the five nearest actin molecules.

A simplified algorithm is used here to calculate the isosurface. In a cylindrical coordinate system {}, we let the axis of the coordinate system coincide with the helical axis of the F-actin filament. The vertices of the isosurface is determined by searching values for all that have positive surface normals. In order to make an efficient rendering, surface coordinates of F-actin are written into a data file and the distortions on the shape of actin monomer caused by the binding of myosin head is ignored.

Similar approach is also used to create the three dimensional shape of myosin subfragment 1 based on the recent structural data. The vertices data of the myosin head is recorded. Myosin heads are placed to the binding sites on actin monomers by applying translations and rotations on the recorded vertices data. A graphic representation of the binding of myosin S1 to F-actin is shown in Fig. 4.

Tropomyosin filament and troponin unit

A tropomyosin molecule is about 41 nm long and is almost a fully -helical coiled coil. Based on the X-ray results, together with analysis of amino acid sequence periodicities, more detailed structural information can be inferred. Tropomyosin molecules form head-to-tail connections and each tropomyosin molecule appears to display a set of discreted binding sites that permit weak linkages of the flexible tropomyosin filament to F-actin along the long-pitch helical strands. A recent three-dimensional reconstruction of thin filament electron micrographs confirms that a movement of tropomyosin on thin filament occurs upon the binding of to TnC.

In our graphic representation, tropomyosin is approximated by flexible coiled tubes. Each tropomyosin molecule cover seven actin sites (Fig. 5). The bind of a myosin head to an actin site can change the tropomyosin position locally around the actin site as well as that around the first-, second-, and third-nearest neighboring actin sites (Fig. 5). The exact position of the tropomyosin at a certain actin site is determined by the configurations of the actin site, the first-, second- and third-nearest neighbors of the actin site, and the nearest troponin molecule.

Troponin, a -sensitive complex, has three subunits: TnC, TnI and TnT. TnT binds one troponin complex to each tropomyosin molecule at intervals 38 nm along F-actin. The troponin complex has an elongated shape with TnC and TnI forming a globular ``head'' region and TnT a long ``tail''. A X-ray structure study indicated that the amino-terminal tail end of TnT spans the head-to-tail joint of tropomyosin filaments, and that the head region of the troponin complex binds about 20 nm away near residues 150-180 of the tropomyosin molecule. Although many aspects about the structure of troponin and its binding to actin remain ambiguous, a simple graphic representation is proposed (Fig. 6). As shown in Fig. 6, there are two stable states for each troponin molecule: one corresponds to -bound TnC and the other corresponds to non--bound TnC.

Dynamics of the thin filament

Putting the above graphic representations together, we can see that the states of the automata array at any given time step can be directly mapped into the geometric shapes of protein configurations (Fig. 7). A dynamic simulation can thus be viewed in a form of video images. This allows us to investigate various processes on thin filaments not only in temporal and spatial dimensions, but also from the structural perspective. For example, the dynamic simulation of thin filaments shown in Fig. 3 can be represented as a three-dimensional graphic animation.

Fig. 8 shows a sequence of video images corresponding to the evolution of the automata array from time step 110 to 121 in (Fig. 3). The length of the time step in the simulation is 1 to 10 ms [1].

The diffusion processes of myosin heads and particles in the solution as seen in Fig. 8 are simulated by an three-dimensional cellular automaton with a set of collision and movement rules that will be discussed in separate papers. The simulation of the diffusion of myosin heads and calcium particles was coupled with the simulation of thin filaments through an interface control and displayed by the animation program.