The cell cycle switch computes approximate majority

Abstract

Both computational and biological systems have to make decisions about switching from one state to another. The 'Approximate Majority' computational algorithm provides the asymptotically fastest way to reach a common decision by all members of a population between two possible outcomes, where the decision approximately matches the initial relative majority. The network that regulates the mitotic entry of the cell-cycle in eukaryotes also makes a decision before it induces early mitotic processes. Here we show that the switch from inactive to active forms of the mitosis promoting Cyclin Dependent Kinases is driven by a system that is related to both the structure and the dynamics of the Approximate Majority computation. We investigate the behavior of these two switches by deterministic, stochastic and probabilistic methods and show that the steady states and temporal dynamics of the two systems are similar and they are exchangeable as components of oscillatory networks.

Figure 1. Switch kinetics in absence of external control.

We study the convergence time (time to reach a stable state) of switching networks; in each column, one of two stable states can be reached from the initial conditions. All the reaction rates are set equal and all the species start at equal quantities (details are provided in Supplementary Materials). Columns A-D concern the DC, AM, SC, CC switches respectively. Row 1 gives a depiction of the networks from which one can precisely recover the chemical reaction networks according to our notation. A catalytic reaction is represented by a circle on top of an arrow; for example, the top right of (F) contains the reaction b+z→z+y with catalyst z. A reaction like x+y→y+y is said to be autocatalytic. The full diagram (F) depicts two catalysts, z and r, acting on species x and y through a shared intermediary b, representing the 4 reactions x+z→z+b, b+z→z+y, y+r→r+b, and b+r→r+x. We use a more compact pinched-arrow graphical notation (E) to represent the same network as (F), hiding the intermediary species b that is assumed not to enter any other reaction. Note that if multiple catalysts act on the same pinched arrow (as in Fig. 2), they all act on the hidden intermediary in the same way. Row 2 contains deterministic simulations for the mass action ODEs of the respective systems for four values of the initial discrepancy between x and y, from 10% to 0.01% (not meaningful for DC as the system would rest at its initial setting), (Y-axis scale on the right). Row 3 shows sample stochastic simulations consisting of individual traces (black lines) of the Gillespie algorithm. The background heat maps, shown in logarithmic scale, give the probability Pr(x_i|t_j) that a system will have x_i molecules of x at time t_j. The horizontal axis is time, and the vertical axis is concentration (row 2) or number of molecules (row 3). Subscripts ‘s’ are for stochastic variables and ‘p’ for probabilistic variables.

Figure 2. Switch equilibrium in presence of external control.

We study the steady state response of our switching networks. Row 1 shows the four systems of figure 1 extended with external controls sx (switch-to-x) and sy (switch-to-y) in gray. On row 2, the horizontal axis represents the value of the sx input, and the vertical axis the value of the resulting x output at steady state. The value of sy remains fixed throughout; as in figure 1 all other initial values and rates are set equal. In each plot, the black line is the bifurcation diagram for the mass action ODEs of the systems (solid = stable, dashed = unstable steady states), showing that each system except the first one exhibits hysteresis; the axes represent concentrations with values that match the ovelayed plots (arbitrary units). The heat map in the background details the discrete probability distribution (in log scale) of a system composed of 5 molecules of each species, except for sy = 3 and with sx varying in 0..15. A point (sx_i, x_j, z_k) in the heat map, for 0 ≤ z_k ≤ 1, means that at equilibrium (after a sufficiently long time) the discrete probability of the system being found with x_j molecules of species x on an input of sx_i molecules of species sx is equal to z_k. The noisy red line is a single run of a steady-state stochastic simulation with maximum x = 150 and with sy = 30; sx is increased slowly to obtain the lower trajectory and the jump up, and is then decreased slowly to obtain the upper trajectory and the jump down.

Figure 3. Switches in the context of oscillators.

We study the three oscillators depicted in column 1: (A) one consisting of two fully connected AM switches; (B) one where two of the connections are replaced by fixed biases; (C) one where we further replace an AM switch with a CC switch. Column 2 (stochastic high-molecular-count simulations) shows oscillations over time for chosen values of r_e/r_i, where r_e is the common rate for the connections between switches (gray lines), while r_i is the common rate for all the reactions within a switch (black lines). Column 3 explores deterministic parameter variation in phase space (x₁ vs. x₂) and bifurcations (r_e/r_i or sx vs. x₁). Column 4 combines a deterministic plot (black line), and a probabilistic low-molecular-count heat map in phase space (x₁ vs. x₂) for a single value of r_e/r_i, from column 2. The values of sx and sy are fixed to 1/3 of the max value of the switching species, and their reactions have rate r_e.

Figure 4. Nonlinearity in switching systems.

We compare the nonlinear dynamics of the AM switch with some alternatives, via sample individual traces obtained form stochastic simulations. We have uniformly chosen rate 1.0 for unimolecular reactions (in (C) and (D)) and 0.001 for bimolecular reactions, with x = y initial conditions (all other species set at zero), and with 15000 max molecule counts as in figure 1. Panel (A) shows the same AM circuit as in figure 1B. Panel (B) shows a variation with two separate intermediates that are processively modified; the upper trajectories converge with a mixture of x and b, hence we plot x+b. Panel (C) shows mutual competition between x and y via dimerization; the upper trajectories converge with a mixture of x and b (the x-dimer), hence we plot x+2b, the total amount of x. Panel (D) shows direct mutual competition between x and y regarded as two enzymes regulated by Michaelis-Menten reactions; we plot x+c, the sum of the enzyme x plus the complex c with its substrate y. Note that x+c is not constant, and similarly for y+b, and hence the system does not operate as a pair of normal enzymatic reactions where those sums are preserved.

Figure 5. Cell cycle switch with Greatwall loop.

We study the GW network: a version of CC from Figure 1D.1 enriched with a feedback loop where x modulates the s and t biases (now unified into the s species). Panel (B) is analogous to Fig 1D.2, and panel (C) is analogous to Fig 1D.4, both showing an improved activation of x. Panel (D) is a comparison between the switching speeds of AM, GW, and CC, showing GW performing better than CC and about as well as AM.