10. Spatiocyte Simulations at Single-Molecule Resolution

We showed an example of E-Cell4 spatial representation.
Next let's simulate the models with more detailed spatial representation called "single molecule resolution".

10.1. Spatiocyte Lattice-based Method

In spatical Gillespie method, we divided the Space into smaller Space, then we diffuse the molecules in the subvolumes. However, we treated the molecules in each subvolume just as the number of the molecules, and the location of the molecules are NOT determinated.

In other words, the spatical resolution of spatical Gillespie method is equal to the side of a subvolume $l$. To improve this resolution, we need to make the size of $l$ small. But in this method the $l$ must be larger than the (at least) 10 times the diameter of molecule $R$.

How can we improve the spatical resolution to the size of the molecule? The answer is the simulation with single-molecule resolution. This method simulate the molecule not with the number of the molecules, but with the spatical reaction diffusion of each molecule.

E-Cell4 has multiple single-molecule resolution method, here we explain about Spatiocyte lattice-based method. Spatiocyte treats each molecule as hard spheres, and diffuses the molecules on hexagonal close-packed lattice.

Spatiocyte has an ID for each molecule and the position of the molecule with single-molecule resolution. For the higher spatial resolution, Spatiocyte has 100 times smaller time-step than spatical Gillespie, because the time scale of diffusion increases with the square of the distance.

Next, let's try the Spatiocyte method.

There is a distinct difference in the second argument for spatiocyte.World. This is called voxel radius. Spatiocyte defines the locations of the molecules with dividing the space with molecule size, and call the minimum unit for this space as Voxel.

In most cases the size of the molecule would be good for voxel radius. In this example, we set 5 $\mathrm{nm}$ as the radius of the molecule in the space with the side 1 $\mathrm{\mu m}$ . It takes more time to simulate, but the result is same with ODE or Gillespie.

10.2. The Diffusion Movement of Single Molecule

Next let's simulate single molecule diffusion to check the resolution.

new_particle method tries to place a particle to a coordinate in spatiocyte.World. It returns the particle's ParticleID (pid). If new_particle fails to place the particle, it returns None instead of a ParticleID. If a particle is already placed in the coordinate, you can NOT place a particle over it.

Particle contains the particle position, species type, radius, and diffusion coefficient. You can inspect the Particle with the particle's ID, pid.

Let's check Particle first.

get_particle method receives a particle ID and returns the ID and particle (of cource the ID are same with the given one). You can inspect the coordinate of the particle as Real3 with position() method. It is hard to directly read the coordinate, here we printed it after converting to tuple. As you can see the tuple coodinate is slightly different from the original position given as a Real3. This is because Spatiocyte can place the molecule only on the lattice. SpatiocyteWorld places the molecule a center position of the nearest lattice for the argument Real3.

You can visualize the coordinate of the molecule with viz.plot_world method, and check the molecule in the center of the World.

And you can use FixedIntervalTrajectoryObserver to track the trajectory of molecular diffusion process.

Here we visualized the trajectory with viz.plot_trajectory method, you can also obtain it as Real3 list with data() method.

data() method returns nested list. First index means the index of the particle. Second index means the index of the Real3. In this case we threw just one particle, so the first result is 1, and next 501 means time-series coordinate of the only one particle (initial coordinate and the coordinates in 1/0.002 = 500 time points).

Also you can obtain the particles in bulk with list_particles_exact method and a Species.

Please remember list_particles_exact method, this method can be used for other World as well as add_molecules method.

10.3 The Diffusion Coefficient and the Second-order Reaction

The models we have addressed contains a second-order reaction. Let's look at the relationship between this second-order reaction and the diffusion coefficient in Spatiocyte.

Although we used faster kinetic constant than before, the result is same. But by contrast with ODE simulation, you can find the difference between them (solid line is spatiocyte, dash line is ode). Is this fault of Spatiocyte? (No) Actually Spatiocyte reaction rate couldn't be faster, while ODE reaction rate can be faster infinitely.

This is caused by the difference between the definition of reaction rate constant in ODE solver and single molecule simulation method. The former is called "macroscopic" or "effective" reaction rate constant, the latter is called "microscopic" or "intrinsic" reaction rate constant.

The "macroscopic" rate represents the reaction rate in mixed molecular state, meanwhile "microscopic" rate represents the reactivity in molecule collision. So in "microscopic" perspective, the first thing molecules need to react is collision. In Spatiocyte, however, you make this "microscopic" rate faster, you can NOT make the actual reaction rate faster than collision rate. This is called "diffusion-limited" condition. This is similar to what the molecules coordinated disproportionately need time to react.

It is known that there is a relationship between this macroscopic rate constant $k_\mathrm{on}$ and microscopic rate constant $k_a$ in 3D space.

$\frac{1}{k_\mathrm{on}}=\frac{1}{k_a}+\frac{1}{4\pi RD_\mathrm{tot}},$

where $R$ is the sum of two molecule's radius in collision, $D_\mathrm{tot}$ is the sum of diffusion coefficients.

In the case of the above Jupyter Notebook cell, $k_D=4\pi RD_\mathrm{tot}$ is almost 0.25 and "microscopic" rate constant is 1.0. So the "macroscopic" rate constant is almost 0.2. (However unless you specify the configuration for Spatiocyte, the second order reaction rate must be slower than $3\sqrt{2} RD$, and the dissociation constant $k_D$ is also $3\sqrt{2} RD$.) The single molecule simulation method can separate molecular "diffusion" and "reaction" in accurate manner contrary to ODE or Gillespie method supposed well mixed system (that is diffusion coefficient is infinite). However if the microscopic rate constant $k_D$ is small enough, the macroscopic rate constant is almost equal to microscopic one (reaction rate-limited).

10.4. The Structure in the Spatiocyte Method

Next we explain a way to create a structure like cell membrane. Although The structure feature in E-Cell4 is still in development, Spatiocyte supports the structure on some level. Let's look a sphere structure as an example.

To restrict the molecular diffusion inside of the sphere, first we create it.

Visualize the state of the World.

The Sphere class first argument is the center of the sphere, and second argument is the radius. Then we created a Species named C and added it inside the Sphere. The structure in the Spatiocyte method is described by filling the space with the Voxel. In the example above, the Voxels in the sphere are occupied with Species named C.

You can see those distribution with viz.plot_world as above. (However, the number of the species is too large to visualize all. So we plot only a part of it, but actually the sphere is fully occupied with the Species.)

Next we create Species moving inside this sphere. To that end we give location attribute to the Species. After that, you just throw-in molecules to the World with add_molecules function.

Now we restricted the trajectories of Species A on the structure of Species C, and add_molecules works like that. As a note, you need to create the structure before add_molecule.

We can use FixedIntervalTrajectoryObserver to check the restriction of the diffusion area.

pid_list is a list of the first 10 ParticleIDs of A molecules. The trajectories are colored by this 10 species. Certainly the trajectories are restricted in the sphere.

10.5 The structure and the reaction

At the end, we explain about molecular translocation among the structures.

A Species without location attribute is not an member of any structures. In the example above, if you do NOT write location attribute with Species A, A is placed outside of the sphere.

Next let's create a planar surface structure. To create a surface, we need to use three Real3, those are original point (origin) and two axis vector (unit0, unit1): ps = PlanarSurface(origin, unit0, unit1).

Suppose Species A on the surface, ps, and a normal Species B.

It might be hard to see them, but actually the Species B are placed only not on a surface. Then how can we make absorbed this Species B to a surface M and synthesize a Species A?

This means that a Species B becomes A when B collides with a structure M. On the other hand, a species A dissociates from the structure, and becomes M and B on as the reverse reaction direction.

Now you can simulate this model with a structure.

In the dissociation from a structure, you can skip to write the structure. Thus, A > B just means the same as A > B + M in the above. But in the binding, you can NOT. Because it is impossible to create A from B with no M around there. By contrast the species A wherever on the sphere M can create the species B. The first order reaction occurs in either presence or absence of the structure. But, in the case of the binding, the second order reaction turns into the first order reaction and the meaning of rate constant also changes if you ignore M in the left-hand side.