Simulation class for regular equations

class oneD_simulation.simulation

Simulator object for regular equations.

Uses finite-difference method of solving PDE defined in Supplementary Modelling

centre_solution()

Given the simulation is performed on Periodic Boundary Conditions, the central location is undefined. For clarity, this function centres the solution such that the maximal E at the final time-step lies at self.num_x/2.

Over-writes self.y_sol and also saves in self.ysolshift as a copy.

del_1d(y, dx)

Central difference discretisation of the Laplacian.

Parameters:

• y – Spatially discretised 1D function (i.e. a np.ndarray of dtype np.float32) on which the function is applied
• dx – Spatial discretisation lengthscale (np.float32)

Returns:

Spatially discretised Laplacian of y

f(y, t)

System of partial differential equations as defined in the Supplementary Modelling, describing the spatio-temporal evolution of E.

Parameters:

• y – Spatially discretised 1D function of E (i.e. a np.ndarray of dtype np.float32) of length self.num_x
• t – Time-point at which to evaluate (np.float32)

Returns:

Time differential of E at each spatial block (np.ndarray of dtype np.float32 and length self.num_x)

f_apical_on(y, t)

System of partial differential equations as defined in the Supplementary Modelling, describing the spatio-temporal evolution of E modelling cell-cell contacts, where E can load only on the apical membrane.

Parameters:

• y – Spatially discretised 1D function of E (i.e. a np.ndarray of dtype np.float32) of length self.num_x
• t – Time-point at which to evaluate (np.float32)

Returns:

Time differential of E at each spatial block (np.ndarray of dtype np.float32 and length self.num_x)

find_peaks(y)

Count the number of peaks in a solution.

Parameters:y – 1D array of concentrations (i.e. E at a given time t) (np.ndarray of size (self.num_x x 1) and dtype np.float32) Returns:Number of peaks (np.int32)

get_amount()

Find self.amount = int_0^L {E} dx

Returns:self.amount

get_apical_solution()

Crop the solution down to just the apical membrane (as defined by self.l_apical versus self.L).

The solution is first centred before cropping.

get_peaks()

Find the number of peaks for all time-points.

Number of peaks = 0 when the relative height is too low (i.e. discounting spurious multi-peak solutions, where variations are miniscule). The threshold is set by self.pol_thresh.

Returns:Number of peaks self.peaks a np.ndarray of size (n_t x 1) and dtype np.int32

get_rel_height()

Find ∆E, the difference between the maximum and minimum value of E at the final time-step.

Returns:∆E, self.rel_height (a np.float32)

get_rel_heights()

Find ∆E for all time-points

Returns:self.rel_heights (**np.ndarray of shape (n_t x 1) and dtype np.float32)

make_directory(dir)

Makes a new directory, specified by the string dir

Parameters:dir – Directory name (string)

norm(x)

Generic function to normalize a 1D np.ndarray, calculating the normalized array y as:

y = (x-min(x))/(max(x) - min(x))

Parameters:x – A 1D np.ndarray to be normalized. Returns:Normalized np.ndarray, y.

plot_time_series(cmap, show=True, filename=False, apical=False, ylim=(0, 150))

Plot time-series of the simulation, as an overlayed set of lines.

The number of time-points that are sampled is set by the parameter cmap, a (n_sample x 4) np.ndarray of RGBA colour points. cmap can be generated using plt.cm.Reds(np.linspace(0,1,n_sample)) for example.

Parameters:

• cmap – The colormap used to plot the solution (a np.ndarray)
• show – If True, then show the plot.
• filename – str defining the file-name of the plot if is being saved. If False then the plot is not saved.
• apical – Determines whether to plot the whole membrane (False) or just the apical membrane (True) (np.bool)
• ylim – Axis limits on the y-axis (i.e. the concentration of E) (tuple)

set_initial(mean=False, SD=False, override=False, apical_on=False)

Set the initial conditions of E.

If override is False, then the initial condition of E, self.y0, is given by a normal distribution, with mean mean and standard deviation SD

If overide is not False but instead a 1D array of length (self.num_x), then self.y0 is defined as override

If apical_on is True, then outside the apical membrane, self.y0 is set to 1e-17 (<<1)

Parameters:

• mean – Mean value used in normal distribution (np.float32)
• SD – Standard deviation used in normal distribution (np.float32)
• override – Override 1D array if provided (np.ndarray of dtype np.float32), else False
• apical_on – Sets self.y0 to (approx.) 0 outside the apical membrane.

Returns:

set_num_x(n, apical_on=False)

Sets spatial discretisation of number num_x

if apical_on is True, then n specifies the number of spatial blocks within the apical domain (of length self.l_apical), and self.num_x is computed with respect to the ratio of self.l_apical and self.L (the full domain length i.e. cell perimeter)

if apical_on is False, then n defines self.num_x

Parameters:

• n – Number of spatial blocks (np.int32)
• apical_on – Specifies whether apical membrane is considered specifically or not (np.bool)

Returns:

set_t_span(dt, tfin)

Sets temporal discretisation. Saves a 1D array of time-points to self.t_span

Parameters:

• dt – Time-step (np.float32)
• tfin – Final time-step (np.float32)

Returns:

solve(apical_on=False)

Perform the simulation. Uses the scipy.integrate.odeint package to integrate the defined system of PDEs.

If apical_on is True, then simulate with cell-cell contacts (where loading is restricted to the apical membrane)

If apical_on is False, then simulate without cell-cell contacts.

Parameters:apical_on – Specifies whether to simulate with or without cell-cell contacts (np.bool) Returns:self.y_sol, a 2D np.ndarray array (n_t x self.num_x), where n_t is the number of time-steps (as defined by self.t_span).