Simulation class for dimensionless equations

class non_dimensional.simulation

Simulator object for dimensionless form of equations.

Uses operator-splitting Fourier transform 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.

diffuse(X, dt, coeffs)

Perform the diffusion operation, given the spatial distribution of the function being integrated X (“e” here)

Parameters:

• X – Function to be integrated (np.ndarray of dtype np.float32 and size (self.num_x x 1)
• dt – Temporal discretization (Strictly “ds” in the Supplementary Modeling)
• coeffs – Coefficients for the Fourier transform, as supplied by the self.make_coeffs function

Returns:

Spatial distribution of the function after integration by 1 time-step (by amount dt)

f(e, t)

Reaction function, describing the transport terms.

Parameters:

• e – 1D np.ndarray of dtype np.float32 and size (self.num_x x 1) describing the spatially discretized distribution of e
• t – Time-point

Returns:

de/dt – 1D np.ndarray of dtype np.float32 and size (self.num_x x 1)

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_polarisation_timescale()

Find the relative polarization time-scale, the amount of (dimensionless) time until the simulation surpasses a critical polarity threshold (defined by self.pol_thresh)

Returns:self.polarisation_timescale

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

make_coeffs(D)

Used in the operator-splitting Fourier method for simulation. Details provided in Supplementary Modeling Appendix IV

Parameters:D – Diffusion coefficient (**np.float32) Returns:Coefficients for the operator-splitting Fourier method of simulation

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)

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)

react(e, dt)

Perform one iteration of the reaction operation.

Parameters:

• e – 1D np.ndarray of dtype np.float32 and size (self.num_x x 1) describing the spatially discretized distribution of e
• dt – Temporal discretization (Strictly “ds” in the Supplementary Modeling)

Returns:

Spatial distribution of the function after integration by 1 time-step (by amount dt)

react_and_diffuse(e, dt, coeffs)

Perform one complete iteration of the integration, both reacting and diffusing (see Supplementary Modeling Appendix IV)

Parameters:

• e – 1D np.ndarray of dtype np.float32 and size (self.num_x x 1) describing the spatially discretized distribution of e
• dt – Temporal discretization (Strictly “ds” in the Supplementary Modeling)
• coeffs – Coefficients for the Fourier transform, as supplied by the self.make_coeffs function

Returns:

Spatial distribution of the function after integration by 1 time-step (by amount dt)

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

Set the initial conditions of e in the dimensionless equations.

e is measured with respect to the saturation threshold E_{crit}: e = E/E_{crit}

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 for dimensionless equations

Lengthscales are measured with respect to the circumference of the cell L = 100microns.

y = x/L

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 for the dimensionless equations.

Time is measured with respect to k_{off}. s = k_{off} * t

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

Parameters:

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

Returns:

simulate()

Perform full simulation, given parameters, using operator-splitting Fourier method

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).