The simulation script described in this chapter is available at STEPS_Example repository.

This chapter introduces how to model and simulate diffusion systems.
First we will look at how to describe the diffusive motion of molecules
by using object of class steps.model.Diff, then how to import
a tetrahedral mesh by using the steps.utilities.meshio methods
and finally how to create a steps.solver.Tetexact object to be used for
the simulation itself. The `Tetexact`

solver builds on the `Wmdirect`

solver
(we have used up until now) extended for diffusive fluxes between tetrahedral
elements in a mesh. Each individual tetrahedron behaves like a well-mixed
compartment where reactions can take place. Diffusive flux between
tetrahedral elements is represented by a series of first-order reactions
with rate constants derived from the local geometry and the diffusion
constant parameter. Thus, this solver object can be used to simulate
full reaction-diffusion systems in complex geometries, but in this introduction
we will start with a simple diffusion system.

We wish to simulate diffusion of one molecular species from a point source in an infinite volume, a problem chosen for simplicity and with a known analytical solution we can compare to our STEPS simulation results. As the volume in STEPS must of course be finite we will create a large spherical mesh, inject our molecules into the central tetrahedron (as there is no concept of a point source in STEPS) and compare our results to the analytical solution up to a time when there are zero or an insignificant number of boundary events.

To compare our mean results in STEPS to an analytical solution we must solve
the diffusion equation for one spatial dimension, the radial distance from the
point source. The problem is simplified to one dimension because the symmetry
of the problem dictates that, in the deterministic limit, the concentration at any given radial distance r
from the point source will be equal at all points in space forming a
two-dimensional “shell” at that *r*.

If all molecules exist at a single point at time 0, within an infinite boundary, the analytical solution is (see e.g. Crank, J. (1975) The Mathematics of Diffusion. Oxford: Clarendon Press):

\begin{equation} C(r,t)=\frac{N}{8(\pi Dt)^{3/2}}\exp\left(\frac{-r^{2}}{4Dt}\right) \end{equation}where *C* is the concentration (if length units are meters the units of C are number of molecules/ $m^{\text{3}}$)
at radial distance *r* from source at time *t*, *N* is the total number of injected
molecules and *D* is the diffusion constant (in units $m^{\text{2}}/s$) .

To set up our model and run our simulation we will create a Python script, organising the script according to a certain template chosen for a good organisation of our workflow. However, for clarity, in these examples we will show the code as if it was typed at the Python prompt. As in previous chapters we will go through the code step by step and look at the statements in detail as we go.

The first thing to do is to write statements to import all our steps packages
with all the methods available to describe our model and run our simulation.
We will also import other packages we require at this point in the script,
such as `math`

, `numpy`

, `pylab`

and `random`

. We will make use of the `random`

package
to help with selecting tetrahedrons from our mesh to sample (which we look at
in detail later) and `math`

contains many useful basic mathematical functions we'll use
for finding the analytical solution:

```
In [1]:
```import math
import numpy
import pylab
import random
import time
import steps.model as smodel
import steps.solver as solvmod
import steps.geom as stetmesh
import steps.rng as srng

```
In [2]:
```# The number of iterations to run
NITER = 10
# The data collection time increment (s)
DT = 0.001
# The simulation endtime (s)
INT = 0.101
# The number of molecules to be injected into the centre
NINJECT = 10000
# The number of tetrahedral elements to sample data from.
SAMPLE = 2000
# The diffusion constant for our diffusing species (m^2/s)
DCST = 20.0e-12

At what stage these constants will be used will become clear as we work through the code.

We want to sample data from individual tetrahedrons so we can analyse spatial
data, so we now create two objects to store the indices and radial distance from
center of the mesh tetrahedrons we will sample. **Note**: *Tetrahedrons are identified by an integer index (as are nodes and
triangles). We look at this in more detail in the geometry section.*

We have decided in this case that we don't want to save data for every single tetrahedron, but rather randomly select 2000 tetrahedrons by setting the SAMPLE variable to 2000, perhaps due to memory constraints. We will look at how we select which tetrahedrons to sample in Geometry specification , but for now we just create NumPy arrays initialized to zeros. The reason for creating these arrays at this point in the script will become clear later:

```
In [3]:
```# Array to hold tetrahedron indices (integers)
tetidxs = numpy.zeros(SAMPLE, dtype='int')
# Array to hold tetrahedron radial distances (floats)
tetrads = numpy.zeros(SAMPLE)

So we now move on to our model description. This time we will organise the code
into a function, which will return the steps.model.Model object we create. It is
entirely up to you if you wish to organise your model description in this way,
but it can be useful for larger models. **Note**: *In this way, for example, multiple model descriptions can be defined
in a separate module with each description clearly separated inside functions.
You can then import whichever model description objects you chose into the
simulation scripts. We will not explore this topic in detail here, but it is a
good idea to keep in mind that this organisation is an option.*

This is our first function
definition, so lets mention a little about the syntax for defining functions in
Python. Firstly, we use the `def`

statement to create a function object and assign
it a name. Then we must provide all our function code with the same indentation.
As soon as our indentation returns to the indentation for the `def`

statement, we
exit the function definition. We wish to return our steps.model.Model object,
so we will provide a `return`

statement at the end of the function. First we create
our function and name it `gen_model`

. In this simple example the function will not
require any arguments, so its signature will be `gen_model()`

.

We start, as always, by creating our steps.model.Model container object `mdl`

.

```
In [4]:
```mdl = smodel.Model()

`A`

) and our volume system `vsys`

, much as in previous chapters.

```
In [5]:
```A = smodel.Spec('A', mdl)
vsys = smodel.Volsys('cytosolv', mdl)

After that we can create our diffusion rule. In STEPS this means creating a steps.model.Diff object and assigning it to a volume system (steps.model.Volsys). As well as the usual identifier string and a reference to the parent volume system, a required parameter for the object construction is a reference to the molecular species object to which this diffusion rule applies. An optional parameter to the object constructor is the diffusion constant, which is given in s.i. units (i.e. $m^{2}/s$ so for example a diffusion constant of $100 \mu m^{2}/s = 100*10^{-12} m^{\text{2}}/s$).

The default value for the diffusion constant can be changed with object method steps.model.Diff.setDcst and can even be changed from the default value in any compartment during simulation, much like the reaction constants we looked at in previous chapters. However, in this model we will not alter the diffusion constant later in the script, so this is the value that will be used during our simulation. Recall we defined the diffusion constant at the beginning or our script, the reason for which is that this value is available in a variable when we come to finding the analytical solution.

```
In [6]:
```diff_A = smodel.Diff('diff_A', vsys, A)
diff_A.setDcst(DCST)

So our complete function is:

```
In [7]:
```def gen_model():
mdl = smodel.Model()
A = smodel.Spec('A', mdl)
vsys = smodel.Volsys('cytosolv', mdl)
diff_A = smodel.Diff('diff_A', vsys, A)
diff_A.setDcst(DCST)
return mdl

We now move on to describing our geometry. This is the section that stands out as very different to our previous well-mixed simulations. The methods we provide for describing mesh-based geometry in STEPS give tools for importing meshes from some powerful mesh-generation packages, and methods for initialising and controlling your simulation conditions which are beyond the ground covered in this chapter. It is likely that you will only start to get the feel of how to use these methods to achieve your required description in more complex models with hands-on experience. This chapter begins to introduce some useful methods for mesh manipulation and it is hoped that this will give the user enough experience with the workflow to be able to go on to more advanced object manipulation that may be required for more complex models. A full list of tetrahedral mesh methods is available in steps.geom.Tetmesh and steps.utilities.meshio.

We chose to structure all our geometry code into a function, much like our model description.

`sphere_rad10_11Ktets.xml`

and `sphere_rad10_11Ktets.txt`

available in folder `meshes/`

in the current working
directory, which we created previously with steps.utilities.meshio.saveMesh from a mesh we imported from
a mesh-generator with the steps.utilities.meshio.importAbaqus function.

```
In [8]:
```import steps.utilities.meshio as smeshio
mesh = smeshio.loadMesh('meshes/sphere_rad10_11Ktets')[0]

Our tetrahedral mesh geometry object is very different from our well-mixed geometry
(steps.geom.Geom) object. The mesh geometry is described by a
steps.geom.Tetmesh object, which contains all the functionality of a
steps.geom.Geom object, extended with many more methods which only make sense
for a tetrahedral mesh. A steps.geom.Tetmesh object is created in steps.utilities.meshio.loadMesh
and returned to the caller, so in the above code the object is referenced by
variable `mesh`

. We will be introduced to some of the steps.geom.Tetmesh methods as we find our
sample tetrahedrons, but first we must create our mesh compartments. A compartment
object in a mesh is of type steps.geom.TmComp and requires a little extra
information than a well-mixed compartment.

A tetrahedral-mesh compartment is comprised of a group of tetrahedrons, so we must supply the object constructor
with the indices of the enclosed tetrahedrons in a Python sequence (e.g. a list).
A mesh can be separated into as many compartments as the user wishes,
though compartments should usually be separated physically by a boundary
(i.e. by a membrane) as *there is no implicit diffusion between compartments
in STEPS, unless connected by a Diffusion Boundary (see* Diffusion Boundary *)*.

If a user wishes to modify behaviour for certain sections of a compartment, this can be achieved by grouping
tetrahedrons together and utilising the simulation methods, all achievable in
the Python interface (see steps.solver.Tetexact for all available methods). However,
for our example we only wish to have one compartment, and for that compartment
to enclose the entire mesh. So we use a steps.geom.Tetmesh object method to
return the number of tetrahedrons in the mesh, and then pass a sequence of all
the indices to the steps.geom.TmComp object constructor. **Note**: *Tetrahedron indices in STEPS always begin at 0 and increment by 1, regardless of their indices in the mesh-generation software. So if a mesh has n tetrahedrons, we can use the Python function range(n) will return a sequence* `[0, 1, ..., n-1]`

*and iterate through each of the tetrahedrons in the mesh.*

```
In [9]:
```# Find the total number of tetrahedrons in the mesh
ntets = mesh.countTets()
# Create a compartment containing all tetrahedrons
comp = stetmesh.TmComp('cyto', mesh, range(ntets))
comp.addVolsys('cytosolv')

Note that we do not (and indeed can not) set the volume of the compartment because the volume is calculated from the combined volume of the enclosed tetrahedrons.

And that's it for our relatively simple geometry description. The remainder of our `gen_geom()`

function
is used to collect, at random, the tetrahedrons to sample data from. This is
introduced here because it is often undesirable to collect data from all
tetrahedrons in large meshes and the user may wish to pick and chose certain
tetrahedrons to sample. Such groups can be stored in a Python sequence with for
loops used to loop over these groups and set simulation parameters or collect data.
In this simple example we will just store the central tetrahedron and it's 4
neighbours, then find the rest at random, making sure not to store the same
tetrahedron more than once. We will store the sample tetrahedron indices in the
`tetidxs`

NumPy array we created at the top of our script. Along the way we will
be introduced to some new steps.geom.Tetmesh methods, which will be described as
we go along. This section is intended to be only an introduction to finding
information from the mesh, though a full list of the many Tetmesh methods that can
be used for more complex tasks is available in steps.geom.Tetmesh. First, we use the
steps.geom.Tetmesh.findTetByPoint method to get the index of the tetrahedron in the centre of our
mesh. steps.geom.Tetmesh.findTetByPoint returns the tetrahedron by index that encompasses the
location given in Cartesian coordinates (in a Python sequence of length 3), and returns `-1`

if the location
given is not inside the mesh. The mesh is a sphere, radius 10 microns, centered
on the origin, so the centre of the mesh is simply at 0.0, 0.0, 0.0 in Cartesian
coordinates. The returned idx is stored in our `tetidxs`

array:

```
In [10]:
```# Fetch the central tetrahedron index and store:
ctetidx = mesh.findTetByPoint([0.0, 0.0, 0.0])
tetidxs[0] = ctetidx

Next we wish to make sure that we include data from around the central tetrahedron,
so we find the central tetrahedron's four neighbours. To do this we use method
steps.geom.Tetmesh.getTetTetNeighb, which returns any tetrahedron's 4 neighbours by index in a
tuple. If any neighbour index is returned as -1 this means that this face of the
tetrahedron is on the boundary and therefore has no neighbour in that direction. **Note**: *This property can be very useful if you wish to find information
about border tetrahedrons or surface triangles.*

In this example it is safe to assume that the central tetrahedron is not
on a surface and we add our 4 neighbour indices to our `tetidxs`

array:

```
In [11]:
```# Find the central tetrahedron's four neighbours:
neighbidcs = mesh.getTetTetNeighb(ctetidx)
tetidxs[1],tetidxs[2],tetidxs[3],tetidxs[4] = neighbidcs

```
In [12]:
```# Keep track how many tet indices we have stored so far
stored = 5
# Find the maximum and minimum coordinates of the mesh:
max = mesh.getBoundMax()
min = mesh.getBoundMin()
# Run a loop until we have stored all tet indices we require
while (stored < SAMPLE):
# Fetch 3 random numbers between 0 and 1:
rnx = random.random()
rny = random.random()
rnz = random.random()
# Find the related coordinates in the mesh:
xcrd = min[0] + (max[0]-min[0])*rnx
ycrd = min[1] + (max[1]-min[1])*rny
zcrd = min[2] + (max[2]-min[2])*rnz
# Find the tetrahedron that encompasses this point:
tidx = mesh.findTetByPoint([xcrd, ycrd, zcrd])
# -1 was returned if point is outside the mesh
if (tidx == -1): continue
if (tidx not in tetidxs):
tetidxs[stored] = tidx
stored += 1

This example is intended to demonstrate that there is a lot of functionality in STEPS to enable you to find and store whatever spatial information is required that could not be passed on from the mesh generator, and that some knowledge of Python is very useful at this stage to enable you to produce code to get maximum benefit from the available methods. This geometry description stage is a good time to find and collect whatever spatial information is required for simulation initialization and data collection. We should note that in this example there is little error checking and more should be included in real simulation scripts (for example SAMPLE must be lower than the total number of tetrahedrons in the mesh). For a full list of the available methods please see steps.geom.Tetmesh.

Now, the final task we wish to perform at the geometry level
is to find the radial distances of the tetrahedrons and fill our `tetrads`

array
with this information. These are stored separately from the indices in our example (for clarity)
and we must make sure that the distances saved
relate to the distance for the tetrahedron at the same location in the `tetidxs`

array, although we could easily have stored the indices and radial distances
together in a 2D array. We will take the radial distance as the distance from
the tetrahedron's barycenter to the barycenter of the central tetrahedron.
To find the barycenters we use method steps.geom.Tetmesh.getTetBarycenter, which returns the
barycenter Cartesian coordinates in a tuple.

```
In [13]:
```# Find the barycenter of the central tetrahedron
cbaryc = mesh.getTetBarycenter(ctetidx)
for i in range(SAMPLE):
# Fetch the barycenter of the tetrahedron:
baryc = mesh.getTetBarycenter(tetidxs[i])
# Find the radial distance of this tetrahedron to mesh centre:
r = math.sqrt(math.pow((baryc[0]-cbaryc[0]),2) \
+ math.pow((baryc[1]-cbaryc[1]),2) \
+ math.pow((baryc[2]-cbaryc[2]),2))
# Store the radial distance (in microns):
tetrads[i] = r*1.0e6

Finally, we return the steps.geom.Tetmesh object required for simulation object construction.

The complete function then reads as follows.

```
In [14]:
```import steps.utilities.meshio as smeshio
def gen_geom():
print("Loading mesh...")
mesh = smeshio.loadMesh('meshes/sphere_rad10_11Ktets')[0]
print("Mesh Loaded")
# Find the total number of tetrahedrons in the mesh
ntets = mesh.countTets()
# Create a compartment containing all tetrahedrons
comp = stetmesh.TmComp('cyto', mesh, range(ntets))
comp.addVolsys('cytosolv')
print("Finding tetrahedron samples...")
# Fetch the central tetrahedron index and store:
ctetidx = mesh.findTetByPoint([0.0, 0.0, 0.0])
tetidxs[0] = ctetidx
# Find the central tetrahedron's four neighbours:
neighbidcs = mesh.getTetTetNeighb(ctetidx)
tetidxs[1:5] = neighbidcs
# Keep track how many tet indices we have stored so far
stored = 5
# Find the maximum and minimum coordinates of the mesh
max = mesh.getBoundMax()
min = mesh.getBoundMin()
# Run a loop until we have stored all tet indices we require
while stored < SAMPLE:
# Fetch 3 random numbers between 0 and 1
rnx = random.random()
rny = random.random()
rnz = random.random()
# Find the coordinates in the mesh that these numbers relate to
xcrd = min[0] + (max[0] - min[0]) * rnx
ycrd = min[1] + (max[1] - min[1]) * rny
zcrd = min[2] + (max[2] - min[2]) * rnz
# Find the tetrahedron that encompasses this point.
tidx = mesh.findTetByPoint([xcrd, ycrd, zcrd])
# -1 was returned if point is outside the mesh:
if tidx == -1: continue
if tidx not in tetidxs:
tetidxs[stored] = tidx
stored += 1
# Find the barycenter of the central tetrahedron
cbaryc = mesh.getTetBarycenter(ctetidx)
for i in range(SAMPLE):
# Fetch the barycenter of the tetrahedron:
baryc = mesh.getTetBarycenter(int(tetidxs[i]))
# Find the radial distance of this tetrahedron to mesh center:
r = math.sqrt(math.pow((baryc[0] - cbaryc[0]), 2) + \
math.pow((baryc[1] - cbaryc[1]), 2) + \
math.pow((baryc[2] - cbaryc[2]), 2))
# Store the radial distance (in microns):
tetrads[i] = r * 1.0e6
print("Tetrahedron samples found")
return mesh

`Tetexact`

Now it’s time to run a simulation and visualize the collected data, much as we did in previous chapters, though this time collecting and plotting spatial data, that is concentrations from individual tetrahedrons and not whole compartments. This time we must call the `gen_model`

and `gen_geom`

functions to set up our model and return the container objects. We then create our random number generator object just as for the `Wmdirect`

simulations.

```
In [15]:
```model = gen_model()
tmgeom = gen_geom()
rng = srng.create('mt19937', 512)
rng.initialize(2903)

```
```

```
In [16]:
```sim = solvmod.Tetexact(model, tmgeom, rng)

```
In [17]:
```tpnts = numpy.arange(0.0, INT, DT)
# Find how many "time points" we have
ntpnts = tpnts.shape[0]
# Create the data structure: iterations x time points x tet samples
res = numpy.zeros((NITER, ntpnts, SAMPLE))

`gen_geom()`

function, though this is of course an option). We then use solver method steps.solver.Tetexact.setTetCount to set the number of molecules in the central tetrahedron at time `t = 0`

to the number stored in variable `NINJECT`

(default number in all tetrahedrons is zero set by steps.solver.Tetexact.reset). We will then run our simulation and collect the data in a few lines of code in nested for loops:

```
In [18]:
```# Fetch the index of the tetrahedron at the centre of the mesh
ctetidx = tmgeom.findTetByPoint([0.0, 0.0, 0.0])
# Run NITER number of iterations:
for i in range(NITER):
sim.reset()
# Inject all molecules into the central tet:
sim.setTetCount(ctetidx, 'A', NINJECT)
for j in range(ntpnts):
sim.run(tpnts[j])
# Loop over the tetrahedrons we are saving data for
for k in range(SAMPLE):
# Save the concentration in the tetrahedron, in uM
res[i, j, k] = sim.getTetConc(int(tetidxs[k]), 'A') * 1.0e6

That is all the code we require to run our simple diffusion simulation.

Notice that the steps.solver.Tetexact.getTetConc returns the concentration in Molar units. Recall that all units in STEPS are s.i. units *with the exception of concentration, which is Molar units*. We then convert Molar units to micro-Molar by multiplying by $10^6$.

We wish to look at the mean concentration in the tetrahedrons over all our iterations, so we simply use the `numpy.mean`

function as in previous chapters:

```
In [19]:
```res_mean = numpy.mean(res, axis=0)

So now we come to plotting our data. Now that we have spatial information the data we wish to plot is different to our previous well-mixed simulations where we were plotting the concentration in a well-mixed compartment. We will plot the mean concentration from individual tetrahedrons against their radial distance from origin, but at many different time points. To achieve this we will create another function, this time with a parameter relating to the “time point“ we wish to plot. We can then call this function with a “time point“ argument and our function will plot concentration vs radial distance at the time relating to that “time point“, as desired. In our function we also label the axis and title the plot with the time.

```
In [20]:
```%matplotlib inline
import matplotlib.pyplot as plt
def plotres(res_mean, tidx):
if tidx >= INT / DT:
print("Time index is out of range.")
return
plt.scatter(tetrads, res_mean[tidx], s=2)
plt.xlabel('Radial distance of tetrahedron ($\mu$m)')
plt.ylabel('Concentration in tetrahedron ($\mu$M)')
t = tpnts[tidx]
plt.title('Unbounded diffusion. Time: ' + str(t) + 's')
plotanlyt(t)
plt.xlim(0.0, 10.0)
plt.ylim(0.0)

`plotanlyt`

. This function will plot the concentration from the analytical concentration given by our equation. The function for plotting the analytical solution is provided here, but we will not go through this code in detail. Here we can see why the diffusion constant was stored in variable `DCST`

at the top of our script:

```
In [21]:
```def plotanlyt(t):
segs = 100
anlytconc = numpy.zeros(segs)
radialds = numpy.zeros(segs)
maxrad = 0.0
for i in tetrads:
if i > maxrad: maxrad = i
maxrad *= 1e-6
intervals = maxrad / segs
rad = 0.0
for i in range(segs):
# Find the conc from analytical solution, and convert to mol/L
anlytconc[i] = 1.0e3 * (1 / 6.022e23) * \
((NINJECT / (math.pow((4 * math.pi * DCST * t), 1.5))) * \
(math.exp((-1.0 * (rad * rad)) / (4 * DCST * t))))
radialds[i] = rad * 1e6
rad += intervals
plt.plot(radialds, anlytconc, color='red')

And that is everything we need to set up and run our simple diffusion simulation and plot the data, alongside the analytical solution for comparison. With this structure, it is intended that the plotting functions is called interactively, giving us the chance to visualise a number of different time plots, then save whichever plots we chose. It often makes sense to output the data to a file, then write plotting functions in separate modules that can load the saved data from these files and plot. This requires some knowledge of writing and reading files in Python, but like most operations in Python, can usually be picked up quite quickly.

Let’s assume we have contained all of our unbounded diffusion code from this chapter in a Python file diffusion.py (which can be found in examples/tutorial). We can then run our simulation interactively in Python by importing the module, then visualising the data with the plotres function we defined. For this example lets provide a call to our plotres function with argument 100 meaning we will plot data at “time point“ 100 (corresponding to time 0.1s), our last “time point“:

```
In [22]:
```plt.figure(figsize=(10,6))
plotres(res_mean, 100)

```
```

Version 2.3 and later of STEPS include an approximate spatial stochastic solver called `TetOpSplit`

(steps.mpi.solver.TetOpSplit). This solver applies an approximation for diffusion whereas reactions are solved by the SSA. The solver is designed to approximate the exact solver, `Tetexact`

, to high accuracy and may perform significantly faster then Tetexact for some models. A full description of the method is available at: http://arxiv.org/abs/1512.03126. Although the method differs significantly from the exact stochastic solver, usage is very similar with only one different command necessary in this example to simulate the model with `TetOpSplit`

instead of `Tetexact`

. That is, instead of creating a Tetexact solver object we create a TetOpSplit solver object with the following command:

```
In [23]:
```import steps.mpi.solver as mpisolvmod
sim_tos = mpisolvmod.TetOpSplit(model, tmgeom, rng, False, \
[0] * tmgeom.ntets)

```
```

`Tetexact`

, using the reference to the `TetOpSplit`

object `sim_tos,`

writing over the data recorded to `NumPy`

array res and recreating res_mean for use by the plotting function:

```
In [24]:
```# Run NITER number of iterations:
for i in range(NITER):
sim_tos.reset()
# Inject all molecules into the central tet:
sim_tos.setTetCount(ctetidx, 'A', NINJECT)
for j in range(ntpnts):
sim_tos.run(tpnts[j])
# Loop over the tetrahedrons we are saving data for
for k in range(SAMPLE):
# Save the concentration in the tetrahedron, in uM
res[i, j, k] = sim_tos.getTetConc(int(tetidxs[k]), 'A') * 1.0e6
res_mean = numpy.mean(res, axis=0)

We can now plot the results obtained with `TetOpSplit`

```
In [25]:
```plt.figure(figsize=(10, 6))
plotres(res_mean, 100)

```
```