Smallest Chemical Reaction System with Hopf Bifurcation

A while back Wilhelm, and Heinrich published a paper that described the smallest chemical network that could display a Hopf bifurcation. That is, the chemical species oscillated. The paper that describes this result is:

Wilhelm, Thomas, and Reinhart Heinrich. “Smallest chemical reaction system with Hopf bifurcation.” Journal of mathematical chemistry 17.1 (1995): 1-14.

This is a diagram of the network taken from their paper:

Here is a Tellurium script that uses Antimony to define the model (Note that $A means that species A is fixed):

The paper gives parameter values that result in oscillations but I wanted to find some other parameter sets. One way to do this is to load the model into the SBW slider control. By changing the sliders one can observe the dynamics. Here I used pathwaydesigner ( to do the same thing. I first exported the model as SBML using:

I loaded the saved SBML file into pathwayDesigner and used the autolayout plugin to get the following network:

I started the slider plugin and varied the sliders until I saw oscillations. In fact, it didn’t take much to get oscillations, all I had to do was increase the value of k1:

I copied the new value of k1 to the Tellurium model (see above) and got the following output:

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C Based Reduce Row Echelon Code

I recently needed some code to compute the reduced row echelon of a matrix. Applications such as Matlab, Mathematics, sympy and R support this functionality out of the box. Libraries such as LAPACK do not, including the linear algebra package in Python. The linear algebra package in Python has some surprising omissions which appear to be by design.

Here I include a C based function that will compute the reduced row echelon. It uses partial pivoting to prevent numerical stability but I don’t know how it would fare when confronted with a large matrix (i.e > 500 rows or columns).

It uses a simple matrix type called TMATRIX. It shouldn’t be difficult to modify code if you use a different matrix structure.

To call rref use:

The 1e-9 argument is the setting that decides whether a value is zero or not. Numbers below tolerance are considered zeros.

The output matrix givne the input should be:

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Plotting Bar graph of Species Concentrations in Tellurium

I had a model with 27 flaoting species and I wanted to plot the steady state concentrations on a histogram where the labels were the names of the different species. Here is a general purpose script that will do that:

The function has a default width and height for the resulting plot. The default widens the usual size so that the labels don’t collide. The first argument is the libroadrunner instance.  To given a contrived example, the following is a model of 8 floating species that form a linear chain governed by simple irreversible mass-action kinetics.

The resulting plot is shown below:


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Multilayered Cascade using TikZ

A summer student working in my lab, Ming Hong Lui from Hong Kong University (HKUST), worked on the perturbation analysis of signaling cascades and in his writeup he use TikZ to draw a nice cascade diagram which I present here.

This code draw the following figure:

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Another Inhibition Pathway Diagram using TikZ

Here is another pathway diagram I needed to draw using TikZ. IN this case I needed an inhibited step. This was more tricky because I needed the inhibition line to point midway to a reaction but without touching the reaction itsef.

To solve this I had to ask stackoverflow and within 2 hours I had an answer. The linear portion of the pathway is straightforward:

The question was how to draw the inhibition line from inhibitior I to reaction v2. The trick is to use the tikz calc extension package which can be used to do more more advanced coordinate calculations.


The line to add that that will draw the vertical inhibition line is:

\draw[|-,line width=1.2pt] ([yshift=4pt]$(S)!.5!(P)$) –++(0,1.5cm)node[above]{$I$};

Let’s decompose this. The first two arguments in the \draw command [[|-,line width=1.2pt] simply set the arrow style (blunt end) and the line width. The second part specifies the coordinates of the line itself and the remainder node[above]{$I$}; indicates what text to draw and where to draw it relative to the line. The real meat is in the drawing line section, that is:

([yshift=4pt]$(S)!.5!(P)$) –++(0,1.5cm)

The basic syntax for a line coordinate is (x1,y1) — (x2,y2). In the example there is a modification to this where the coordinate is specified as (x1,y1) — ++(x2,y2). The ++ means that the coordinate at x2,y2 is computed relative to x1,y1, that is the coordinate at x2,y2 is actually (x1,y1)+(x2,y2). The (0,1.5cm) then means that the coordinate for the end point has the same x coordinate but the y coordinate is displaced upwards by 1.5cm. Note that the tikz y coordinate is like a normal graph plotting axis.

The trickest bit is computing the starting point for the vertical line. Note that this has to be mid point between nodes S and P.  This requires some coordinate calculations. The x,y coordinate is specified by $(S)!.5!(P)$). The bit in front, ([yshift=4pt], just moves the computed y coordinates up 4pts.

The  text inside the $$ means that we are doing a coordinate calculation,ie $….$ represents a math calculation. (S) and (P) represent the coordinates of the nodes S and P respectively. The important bit is !.5!. The explanation point is a pathway modifier and can be put between two coordinates.

(S)!.5!(P) means compute the coordinate that is half way between (S) and (P).

The last thing to add is that I nocticed that the blunt end was a bit too sort. To widen the blunt end I used the arrow.meta extension package. This allows one to modify the size of the arrow heads, in this case I used the following to widen the blunt end to 5mm.


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Drawing a Pathway Fragment with Inhibition using Tikz

Here is another quick pathway fragment I needed today. This won’t scale well because I’ve used some fixed dimensions eg the width of the lines and the size of the text. But these are easily adjusted if you want to size the figure differently. The node distance = 2.5cm gives the sizes for the arrows except for v3. One might be able to calculate the some of the fixed sizes based on the node distance but I didn’t have time to investigate this.



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Drawing a ‘Futile’ Cycle using Tikz

I was in need of a diagram of a futile cycle. I tried illustator but I didn’t have the right LaTeX fonts so the figure did blend well with the rest of the document. I decided to make one using TiKz. And here it is.


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Plotting 3D graphs using Python and Tellurium

As an example I wanted to show how one could plot a 3D phase plot. A great example to use for this is the Lorenz Attractor. This system is interesting because it displays chaotic behavior. The differential equations for the system are given by the following three:

{\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma (y-x),\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=x(\rho -z)-y,\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z.\end{aligned}}

Different values for the parameters, sigma, rho and beta, lead to different behaviors. The Python script below that uses Tellurium plots a 3D phase plot:

# The Lorenz Model

# The Lorenz system of equations is the classic set of
# equations which exhibit chaotic dynamics. The equations
# represent a highly simplified model of 2D thermal
# convection known as Rayleigh-Benard convection;

# The three variables, x, y and z represent the
# convective overturning, the horizontal temperature
# and vertical temperature variation respectively;

## The parameters a, b and c represent approximately
# the energy losses within the fluid due to viscosity,
# the temperature difference between the two plates
# which separate the fluid and the ratio of the vertical
# height of the fluid layer to the horizontal extent
# of the convective rolls respectively;
import tellurium as te
import roadrunner
from mpl_toolkits.mplot3d import Axes3D
import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np

r = te.loada ('''
// Define the model in terms of three differential equations;
$s -> x; -a*(x - y);
$s -> y; -x*z + b*x - y;
$s -> z; x*y - c*z;

a = 10; b = 28; c = 2.67;

# Uncomment/comment the following lines to obtain behaviour you wish to observe;

# Non-chaotic oscillations;
#r.x = 5.43027; r.y = 26.74167; r.z = 46.0098;
#r.a = 4; r.b = 91.9; r.c = 2.67;

# Single oscillation;
#r.x = 32.82341; r.y = -44.84651; r.z = 258.02389;
#r.a = 4; r.b = 191.9; r.x = 2.67;

# Stable steady-state;
#r.x = 0.1; r.y = 0.1; r.z = 0.1;
#r.a = 10; r.b = 10; r.c = 2.67;

# Chaos
r.x = 0.96259; r.y = 2.07272; r.z = 18.65888;

m = r.simulate (0, 40, 3000, ['x', 'y', 'z']);

fig = plt.figure(figsize=(8,8))
ax = fig.gca(projection='3d')

#theta = np.linspace(-4 * np.pi, 4 * np.pi, 100)
ax.plot(m[:,0], m[:,1], m[:,2], label='Lorenz Attractor')


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How do I change the simulation tolerances in Tellurium?

For very complicated and large models it may be necessary to adjust the simulator tolerances in order to get the correct simulation results. Sometimes the simulator will terminate a simulation because it was unable to proceed due to numerical errors. In many cases this is due to a bad model and the user must investgate the model to determine what the issue might be. If the model is assumed to be correct then the other option is to change the simulator tolerances. The current option state of the simulator is obtained using the getInfo call, for example:

There are a variety of tuning parameters that can be changed in the simulator. Of interest are the relative and absolute tolerances, the maximium number of steps and the initial time step.

The smaller the relative tolerance the more accuate the solution, however too small a value will result in either excessive runtimes or more likely rounadoff errors. A relative tolerance of 1E-4 means that errors are controlled to 0.01%. An optimal value is roughly 1E-6. The absoute tolerance is used when a variable gets so small that the relative tolerance don’t make much sense to apply. In these situations the absolute error tolerance is used to control the error. A small value for the absolute tolerance is often desirable, such as 1E-12, we do not recommend going below 1E-15 for either tolerances.

To set the tolerances use the statements:

Another parameter worth changing if the simulations are not working well is to change the initial time step. This is often set by the integrator to be a relatively large value which means that the integrator will try to reduce this value if there are problems. Sometimes it is better to provide a small initial step size to help the integrator get started, for example, 1E-5.

The reader if refered to the CVODE documentation for more details.

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How do I plot phase plots using Tellurium?

Phase plots are a common way to visualize the dynamics of models where time courses are generated and one variable is plotted against the other. For example consider the following model that can show oscillations:

In this model S2 positively activates reaction v2 thus forming a positive feedback loop. The rate equation for v2 include a Hill like coefficient term, S2^h, this determines the strength of the positive feedback. The oscillations originate from an interaction between the positive feedback and a non-obvious negative feedback loop at S1 beteen v1 and v2.

Let us assign suitable parameter values to this model, run a simulation and plot S1 versus S2.

Running this script by clicking the green button in the toolbar yields the following plot:


What if we’d like to investigate how the oscillations are affected by the parmaeters of the model. For example how does the model bahave when we change k1? One way to do this is to plot simulations at different k1 values onto the same plot. In this case however this will create a difficult to read graph. Instead let us create a grid of subplots where each subplot represents a different simulation.

Here we create a 3 by 3 subplot grid, start a loop that changes k1 and each time round the loop it plots the simulation onto one of the subplots. Running this script results in the following output


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