Basins of attraction and times of response

The cell fusion and reprogramming experiments, described below in section 4.3, would lead to a situation where a switch element, previously repressed, is brought to a level comparable to that of another switch element which was already expressed. This corresponds to an initial situation in which two elements are not at their resting value, which could also describe the situation in cells in the process of differentiating. For the models studied here, if 2 switch elements are competing, 3 outcomes are possible: the switch element at the higher concentration completely represses the other, both coexist and reach a non-zero equilibrium at the same value (only an element which started at the higher concentration can end up predominating), or both go to 0 (extinction). Figures 7 to 10 show which equilibrium the system converges to, as a function of the initial state, for different values of the competition parameter $\alpha$ (each domain from which the system converges to the same equilibrium is a "basin of attraction"). If there are 3 switch elements competing, there are more possibilities, as 2 or 3 elements can coexist in the "on" state. Figures 11 and 12 show the basins of attraction of such a switch (the attraction basins belong to the same system, but were split on two figures to prevent the outer ones from obscuring the inner ones).

The speed at which the competition between the switch elements is carried out could be crucial in determining the dynamical properties of differentiation. We thus computed the time it takes for the system to approach its equilibrium, starting from various initial concentrations of the switch elements (that time is colour-coded in Figures 7 to 12). This time becomes dramatically higher when the initial conditions come close to the borders of the basins of attraction (ie when concentrations are near a threshold around which the system converges to two or more different outcomes). The effect becomes even more pronounced when 3, rather than 2, switch elements are competing (Figures 11 and 12).

To show the effect in more detail, individual trajectories were plotted for a 2-dimensional switch (Figures 13 and 14). For cell fusion and reprogramming experiments, the effect on the concentration of switch elements depends on the dynamics of nuclear import and export. Two types of initial conditions were used: two switch elements were given the concentration that one would have at rest, if it was "on" (Figure 13), or two switch elements were given half that concentration (as cytoplasmic concentrations of proteins expressed exclusively in 1 of 2 equally-sized cells should be cut by half upon fusion; Figure 14). In both cases, the concentrations of the two switch elements, even for that which will eventually prevail, initially go down. The effect is more pronounced for higher values of the initial concentrations, and for close initial values of the two concentrations. This could explain the transient extinction of expression of differentiated markers upon cell fusion (see Discussion): if there is sufficient cooperativity downstream of the switch element, its dip could be sufficient to provoke a temporary extinction of expression of the proteins specific to the differentiated state.

Figure: Colour-coded time of convergence (as defined in Appendix D.2), as a function of the initial conditions in $x_1$ and $x_2$. From the initial conditions to the left of the red region, the system converges to $x_2$ on and $x_1$ off, and the opposite for the initial conditions to the right of the red region. Parameters are $\alpha =0.4$ and $\sigma =100$. $x_1$ and $x_2$ range from 0 to 300.

Figure 8: Same as Figure 7, but with a lower value of $\alpha$, giving a large domain of convergence to an equilibrium where $x_1$ and $x_2$ coexist. Domains of convergence are indicated, and are separated by the two yellow lines. Parameters are $\alpha =0.1$ and $\sigma =100$. $x_1$ and $x_2$ range from 0 to 300.

Figure 9: Same as Figure 7, but with a markedly higher value of $\alpha$. There are two main domains of convergence (to one switch element on and the other off), and a third domain of convergence to 0 (complete extinction of the switch), in a region very close to the upper part of the diagonal (for clarity reasons, the region is indicated as larger than it actually is). Parameters are $\alpha =15$ and $\sigma =100$. $x_1$ and $x_2$ range from 0 to 300.

Figure 10: Same as Figure 7, with $\alpha$ close to the threshold above which 0 is the only equilibrium. The region from which the system converges to 0 has expanded. Parameters are $\alpha =24.75$ and $\sigma =100$. $x_1$ and $x_2$ range from 0 to 300.

Figure: Times of convergence as a function of the initial condition, for a 3-dimensional switch. 4 unconnected domains of convergence to the same equilibrium are shown. For visibility, the 3 other domains are shown in Figure 12. Parameters are $\alpha =0.1$ and $\sigma =25$. A rotation movie is available at http://cinquin.org.uk/High-dimensional_switches_and_the_modeling_of_cellular_differentiation/rotating_graphs.html

Figure 12: Domains in which the same switch as in Figure 11 converges to a state with 2 switch elements on. A rotation movie is available at http://cinquin.org.uk/High-dimensional_switches_and_the_modeling_of_cellular_differentiation/rotating_graphs.html

Figure 13: Time evolution of the concentrations of two switch elements ($x_1$ and $x_2$), for the bHLH dimerisation model. The resting concentration when one element is on and the other off is roughly 8. Initial concentrations differ by 0.7 ( $\mathbf{a}$), or 0.1 ( $\mathbf{b}$). Notice the differences in scales. Parameters are $\alpha =50$ and $\sigma =500$.

$$\begin{array}{c} \includegraphics[width=5in]{figure_12a.eps}... ...egraphics[width=5in]{figure_12b.eps} \\ \mathbf{b} \end{array}$$

Figure: Same as Figure 13, but with initial concentrations at roughly half the equilibrium value when one element is on and all others off. Initial concentrations differ by 0.5 ( $\mathbf{a}$), or 0.1 ( $\mathbf{b}$).

$$\begin{array}{c} \includegraphics[width=5in]{figure_13a.eps}... ...graphics[width=5in]{figure_13b.eps} \\ \mathbf{b} \end{array}$$