This is the second example about diffusion. In this one, we will show that diffusion, coupled with degradation and non-uniform synthesis, can result in concentration gradients.

The code for this example is mostly identical to that of the previous example, therefore we will only explain the parts that differ from the previous example.

Preparation



In [1]:

    
%matplotlib notebook

Imports



In [2]:

    
import multicell
import numpy as np

Problem definition

Simulation and tissue structure



In [3]:

    
sim = multicell.simulation_builder.generate_cell_grid_sim(20, 1, 1, 1e-3)

Biological species



In [4]:

    
sim.register_cell_variable("a")

Constants

In this example, we introduce two additional constants. mu_a is the degradation - and turnover - rate of a (common to all cells) and A is a relative synthesis rate of a (20 in the first 10 cells do, 0 in the others).



In [5]:

    
A = np.array([20] * 10 + [0] * 10)
sim.set_constants({"D_a": 100., "mu_a": 1., "A": A})

Differential equations

This time, the differential equation governing a is slightly more complex, as it includes a degradation term and a synthesis term.



In [6]:

    
def da_dt(simulation, a, c_a, D_a, mu_a, A, adjacency_matrix, **kwargs):
    return simulation.diffusion(D_a, c_a, adjacency_matrix) - mu_a * a + mu_a * A

sim.set_ODE("a", da_dt)

The equation is designed so that the system has a steady state, and the total amount of a in the system at that steady state is independent of the value of mu_a, the turnover rate. We can therefore safely change the value of mu_a to modify the turnover without affecting the scale of the concentration values, which is useful for visualization purposes.

Initial conditions



In [7]:

    
sim.initialize_cell_variables()

a0 = A # Same as in the previous example

sim.set_cell_variable("a", a0)

Duration



In [8]:

    
sim.set_duration(4)
sim.set_time_steps(10, "log2")

Rendering



In [9]:

    
sim.register_renderer(multicell.rendering.MatplotlibRenderer, "c_a", {"max_cmap": 20, "view": (90, -90), "axes": False})

Visualization of the initial state



In [10]:

    
sim.renderer.display("c_a")









    














    











    



Time point: 0.0
c_a: from 0.0 to 20.0114678814

Simulation



In [11]:

    
sim.simulate()









    














    











    



Time point: 0.0078125
c_a: from 1.18271777959e-07 to 19.9998196838






    














    











    



Time point: 0.015625
c_a: from 3.40689200503e-05 to 19.9945245187






    














    











    



Time point: 0.03125
c_a: from 0.00368627258512 to 19.9893159191






    














    











    



Time point: 0.0625
c_a: from 0.106806599049 to 19.8882833579






    














    











    



Time point: 0.125
c_a: from 0.849797545315 to 19.1472791957






    














    











    



Time point: 0.25
c_a: from 2.69338014729 to 17.305347038






    














    











    



Time point: 0.5
c_a: from 4.88245197264 to 15.1171138453






    














    











    



Time point: 1.0
c_a: from 6.19930035089 to 13.8005401514






    














    











    



Time point: 2.0
c_a: from 6.47387224614 to 13.5260191081






    














    











    



Time point: 4.0
c_a: from 6.48275714554 to 13.5171358502

In this example, concentrations are no longer uniform at the steady state, as a result of the turnover of species a.

The competition between diffusion and turnover leads to the formation of concentration gradients around the source of the species. The steepness of the gradient depends on diffusion and turnover rates, relative to each other. The faster the diffusion and the slower the turnover, the smoother the gradient.

Changing the values of D_a and mu_a will affect the steady state. Increasing values of mu_a will increase the turnover rate, while increasing the value of D_a will increase the diffusion rate.