Generic Cell Cycle Model
Introduction to bifurcation theory
1. Vector fields
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Bifurcations can help us to understand how the system reacts for parameter changes: A set of differential equations with a parameter set determine how the system will move in time from an initial condition, how the concentrations of the different regulators change. The direction and speed of this change can be represented by a vector in a multidimensional (as many variables we have) phase space. Each point of the phase space has a vector which tells where the system will move in the next time step (Fig A).
A numerical simulation basically jumps from vector to vector, following the vector field (the multidimensional space with vectors from each point). Special points in the vector field are those which have a zero vector, so the system is not moving from there, it is in steady state. If the system goes back to this point after a small perturbation then it is a stable steady states, if it moves away then it is an unstable steady state. There can be other points in the vector field which has vectors that create a loop and the system oscillates on this periodic orbit (Fig C). The stability of this limit cycle can be determined same way as for points. These special structures and points of the vector fields are called attractors as they attract the solutions from any initial conditions. Each of the attractors has a domain of attraction which consist all points of the vector field from where the system will go to that attractor.As parameters of the system are changing the number and stability of attractors also can change, this change in attractors called bifurcation.
At a bifurcation the system can loose a stable attractor, while at the same point new, different attractors can appear. In the case of cell cycle regulatory system, we can connect different cell cycle phases to different attractors, and transitions between cell cycle phases to bifurcations of the system. Bifurcation theory can tell how the attractors of a system change for parameter modifications, so it can be used to understand how the transitions (bifurcations) between cell cycle phases (attractors) depend on certain parameters.
2. One dimensional bifurcation diagrams
As we defined bifurcations are transitions when number or stability of attractors in a system changes. To visualize these on a bifurcation diagram one should pick an indicator of the system as bifurcation variable and a bifurcation parameter that can change the systems behavior. From this point we can treat our regulatory network as a black box and just check what respond our bifurcation variable gives for bifurcation parameter changes. So we can interpret this diagram as a signal response curve where we plot how the system responds for a change in one of the crucial signals. The respond is measured by our bifurcation variable the signal is changed by the bifurcation parameter (Fig D). We have to find a representative indicator and an important signal of the system to draw a meaningful bifurcation diagram.
What can we read from a bifurcation diagram? Solid and dashed curves are showing how the stable and unstable steady state values of the bifurcation variable develop for change in the bifurcation parameter (Fig D). Bifurcation happens at a certain parameter value, when the number or the stability of steady states changes. The most relevant bifurcation types are collected in Table I at the bottom of this page. At saddle-node (SN) bifurcation, a stable (node) and an unstable steady state (saddle) meet and destroy each other. This is the transition between the phase spaces represented in Fig A and Fig B. This bifurcation can lead to abrupt changes in the bifurcation variable (Fig D), because as one stable state has lost the system has to find a steady state which might be far away from this earlier point (Fig B). This huge change in the systems response at a SN bifurcation resembles the behavior of a toggle switch, when a signal increase has minor effect until the signal strength reaches a threshold (bifurcation point) but after passing it the system turns ON Tyson et al., 2003 . This can be explained by the bistability (existence of two possible stable steady states) between two SN bifurcations Sha et al., 2003 and the transition between these two states at the SN points. Hopf bifurcations change a steady state to a limit cycle and a steady state of opposite stability. If at a bifurcation point a stable steady state changes into stable limit cycle with increasing amplitudes and an unstable steady state, then we talk about a supercritical Hopf bifurcation (Fig D). This is the transition between phase space in Fig B and Fig C.
Fig D. The transitions (bifurcations) between the phase spaces of panel A-C are represented on a one dimensional bifurcation diagram. A possible simulation for the increase in signal is overlaid with light grey. SN and Hopf signs show positions of bifurcation points (see table I for explanation of these and other bifurcation types), bi-stabil region between SN points is noted by gray backgorund.
Bifurcation from an unstable steady state to stable steady state and an unstable limit cycle called subcritical Hopf bifurcation. In this case very often a cyclic-fold (CF) bifurcation changes the stability of the limit cycle, so a stable oscillation surrounds the unstable one (Table 1). This causes that if the system reaches from a stable steady state a HBsub point then large amplitude oscillation appears. Also large amplitude oscillation appears at a SNIPER point, which is a special SN bifurcation, when after the collision of the saddle and the node the remaining steady state is unstable, surrounded with a stable limit cycle. Saddle-loop (SL) bifurcation originates from an unstable steady state (saddle) and also leads to large amplitude oscillations. When this point moves until the end of a saddle and reaches a SN point then a SNIPER bifurcation is created, this bifurcation from SN+SL to SNIPER is called saddle-node loop bifurcation. This example already tells how the so called "co-dimension one" bifurcations can turn into each other at "co-dimension two" bifurcations (Table 2).
Table 1.
3. Two dimensional bifurcation diagrams
The co-dimension two bifurcations of bifurcations can be analyzed on two dimensional bifurcation diagrams, where we show how the positions and types of bifurcations develop for a change in a second parameter. This second parameter can represent the genetic information inside the cells, so we can have a plot that shows how the cell physiology (mass) and genetics (activity of a regulator molecule) determine how and where the cell cycle checkpoints related transitions (bifurcations) occur. Mutations change the activities of molecules so the two dimensional bifurcation diagrams can be used to analyze transitions from wild-type to mutant cell cycle regulation. A one dimensional cut of this diagram is the one dimensional wild-type bifurcation diagram, and another cut is the one dimensional bifurcation diagram of the corresponding mutant. We can visualize this way how crucial cell cycle transitions (co-dimension one bifurcations) are shifted to different cell mass and the transition behaviors are changed at co-dimension two bifurcations in different mutants. Further example of co-dimension two bifurcations (Table 1) is the degenerate Hopf point where the two types of Hopf bifurcation can turn into each other with the help of a CF bifurcation. At a Takens-Bogdanov point a Hopf and a SL bifurcation reach a SN point where the oscillation disappears and the node changes its stability. Two SN bifurcations can define the border of a parameter regime where two stable steady states exist. This bistability disappears in a CUSP bifurcation, when the two SN meet and destroy each other and a single steady state remains.
Table 2.
Here we show a simple hypothetical two dimensional bifurcation diagram with examples of all of the above and one dimensional cuts after each co-dimension two bifurcations to help better understanding of the co-dimension two transitions.
Fig E. Example of a two dimensional bifurcation diagram with one dimensional
cuts.Hypothetical bifurcations of a possible cell cycle regulatory network. The typical bifurcation types are connected to each other in a consequent fashion. Click on numbers 1-6 for one dimensional bifurcation diagram cuts! Click here to see all the six diagrams in one window. |
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If you want to learn more about bifurcation theory we suggest you to look at these books: Kuznetsov, 1995 , Strogatz, 1994 , Fall et al., 2002 .
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