Biological applications of bifurcation theory

The ability to make dramatic change in system output is often essential to organism function, and bifurcations are therefore ubiquitous in biological networks such as the switches of the cell cycle.

At the cellular level, components of a network can include a large variety of proteins, many of which differ between organisms.

Network interactions occur when one or more proteins affect the function of another through transcription, translation, translocation, phosphorylation, or other mechanisms.

Therefore, it can be impossible to predict the quantitative behavior of a biological network from knowledge of its organization.

A phase portrait is a qualitative sketch of the differential equation's behavior that shows equilibrium solutions or fixed points and the vector field on the real line.

As a very simple explanation of a bifurcation in a dynamical system, consider an object balanced on top of a vertical beam.

The mass of the object can be thought of as the control parameter, r, and the beam's deflection from the vertical axis is the dynamic variable, x.

For a more rigorous example, consider the dynamical system shown in Figure 2, described by the following equation:

The system's fixed points are represented by where the phase portrait curve crosses the x-axis.

In this case, because the behavior of the system changes significantly when the control parameter r is 0, 0 is a bifurcation point.

Real biological systems are subject to small stochastic variations that introduce error terms into the dynamical equations, and this usually leads to more complex bifurcations simplifying into separate saddle nodes and fixed points.

Note that the saddle node itself in the presence of error simply translates in the x-r plane, with no change in qualitative behavior; this can be proven using the same analysis as presented below.A common simple bifurcation is the transcritical bifurcation, given by

Tracking the x-intercepts in the phase diagram as r changes, there are two fixed point trajectories which intersect at the origin; this is the bifurcation point (intuitively, when the number of x-intercepts in the phase portrait changes).

The error term translates all the phase portraits vertically, downward if h is positive.

In the left half of Figure 6 (x < 0), the black, red, and green fixed points are semistable, unstable, and stable, respectively.

First, find the fixed points in the phase portrait by setting the bifurcation equation to 0:

Next, determine whether the phase portrait curve is increasing or decreasing at the fixed points, which can be assessed by plugging x* into the first derivative of the bifurcation equation.

The results are complicated by the fact that r can be both positive and negative; nonetheless, the conclusions are the same as before regarding the stability of each fixed point.

The blue and red are solid lines in Figure 7 (left), while the black unstable trajectory is the dotted portion along the positive x-axis.

Once again, the phase portraits are translated upward an infinitesimal amount, as shown in Figure 9.Tracking the x-intercepts in the phase diagram as r changes yields the fixed points, which recapitulate the qualitative result from Figure 7 (right).

A linear stability analysis can also be performed here, except using the generalized solution for a cubic equation instead of quadratic.

The process is the same: 1) set the differential equation to zero and find the analytical form of the fixed points x*, 2) plug each x* into the first derivative

Bistability (a special case of multistability) is an important property in many biological systems, often the result of network architecture containing a mix of positive feedback interactions and ultra-sensitive elements.

[2] For instance, this is important in contexts where a cell decides whether to commit to a particular pathway; a non-hysteretic response might switch the system on-and-off rapidly when subject to random thermal fluctuations close to the activation threshold, which can be resource-inefficient.

Networks with bifurcation in their dynamics control many important transitions in the cell cycle.

For instance, egg extracts of Xenopus laevis are driven in and out of mitosis irreversibly by positive feedback in the phosphorylation of Cdc2, a cyclin-dependent kinase.

[3] In population ecology, the dynamics of food web interactions networks can exhibit Hopf bifurcations.

For instance, in an aquatic system consisting of a primary producer, a mineral resource, and an herbivore, researchers found that patterns of equilibrium, cycling, and extinction of populations could be qualitatively described with a simple nonlinear model with a Hopf Bifurcation.

The system exhibits bistable switching between induced and non-induced states.

[5] Similarly, lactose utilization in E. coli as a function of thio-methylgalactoside (a lactose analogue) concentration measured by a GFP-expressing lac promoter exhibits bistability and hysteresis (Figure 10, left and right respectively).

Figure 1. Example of a biological network between genes and proteins that controls entry into S phase .
Figure 2. Saddle-node bifurcation phase portrait, where the control parameter is varied (labelled ε instead of r , but functionally equivalent). As ε decreases, the fixed points come together and annihilate one another; As ε increases, the fixed points appear. dx/dt is denoted as v.
Figure 4. Unperturbed (black) and imperfect (red) transcritical bifurcations, overlaid. u and p are referred to as x and r , respectively, in the rest of the article. As before, solid lines are stable, and dotted unstable.
Figure 5. Ideal transcritical bifurcation phase portraits. Fixed points are marked on the x-axis, each trajectory in a different color. The direction of the arrows indicates which direction they move as r increases. The red point is unstable, and the blue point is unstable. The black dot at the origin is stable for r < 0, and unstable for r > 0.
Figure 6. Imperfect transcritical bifurcation phase portraits. Five values of r are shown, given relative to the two critical points. Note that the y-intercept value is the same as h , or the magnitude of the imperfection. The green and blue points are stable, while the green red and magenta are unstable. The black dots indicate semistable fixed points.
Figure 8. Ideal pitchfork bifurcation phase portraits. Fixed points are marked on the x-axis, each trajectory in a different color. The direction of the arrows indicates which direction they move as r increases. The red and blue points are stable, and there is a third unstable fixed point at the origin, indicated by the black dot. For r = r crit = 0, the black dot also indicates a semi-stable point that appears and splits into the other three trajectories as r increases.
Figure 9. Imperfect pitchfork bifurcation phase portraits. Four different values of r relative to r crit are shown. Note that the y-intercept value is the same as h , or the magnitude of the imperfection. The red and blue points are stable, while the green one (previously hidden at the origin) is unstable. As in Figure 5, the black dot indicates a semi-stable point that appears and splits into the red and green ones as r increases.