Figure 1.
Schematic Representation of Our Cell Model
The model consists of two networks, i.e., a gene regulatory network and a metabolic network. As a schematic example, simple networks consisted of n = 7 genes and m = 6 metabolic substrates are shown. The red arrows in the regulatory network represent activation of expressions, while green lines with blunt ends represent inhibition. The arrows from a gene to itself mean autoregulation of expressions. As a result of these regulatory interactions, the dynamics of expression levels of proteins have multiple attractors. The metabolic reactions, represented by blue arrows, are controlled by expression levels of corresponding proteins. The correspondence between metabolic reactions and gene products (proteins) are shown by the thin black arrows. The regulatory matrix Wij of the presented network takes W21 = W32 = W33 = W45 = W56 = W67 = W77 = 1, W24 = W53 = W57 = −1, and 0 otherwise. The reaction matrix Con(i,j,k) metabolic network takes a value 1 for the elements (1,3,1)(2,3,2)(3,4,3)(6,3,4)(4,5,5)(6,4,6)(5,6,7), and 0 otherwise. The choice of n = 7 in the figure is only for schematic illustration, and in the actual simulation we used much larger networks with n = 20 ∼ 100. In the present paper, we adopt a much larger network with n = 96 genes and m = 32 substrates.
Table 1.
Summary of Basic Requirements for Our Adaptation Mechanism
Figure 2.
Selection of a Higher Growth State by Noise
(A) Time series of protein expressions xi(t). Ten out of 96 protein species are displayed. The vertical axis represents the expression levels of proteins, and the horizontal axis represents time.
(B) Change in growth rate vg observed during the time interval shown in (A). Initially, the growth rate of the cell fluctuates due to the highly stochastic time course of protein expression. After a few short-lived nearly optimal states (c.f. 4,800 ∼ 5,600 time steps), the cell finds a state of protein expression that realizes a high rate of growth. The parameters are θ = 0.5, μ = 10, ρa = ρi = 0.03, ɛ = 0.1, and D = 0.1. In addition, we enhanced the rate of positive autoregulatory paths, i.e., Wii = 1 for i-th gene, so that the regulatory network has multiple attractors. In the simulations, 30% of activating paths are chosen as autoregulatory paths.
Figure 3.
Distribution of Growth Rate and Escape Probability from an Attractor
(A) The distribution of growth rate. Starting from randomly chosen 105 initial conditions, the distribution of growth rates after 105 time steps are computed with and without noise (σ = 0.2).
(B) Relationship between the growth rate vg and the probability to escape an attractor within a certain period of time. The probability is computed by 105 trials starting from randomly chosen initial conditions. After a cell reaches a stable state, noise (σ = 0.2) is added, and the time it takes the cell to escape from the corresponding attractor is measured. The y-axis represents the probability that the cellular state is kicked out of the original state within 103 time steps, and the horizontal axis shows the growth rate vg of the original state.
Figure 4.
Adaptation Process over Several Environmental Conditions
(A) Time series of protein expressions xi(t) when the environmental condition is altered. The environmental conditions, i.e., substrates having nonzero Yi, are changed at time points indicated by arrows.
(B) Change of growth rate vg in the same time interval as (A). After the environmental changes, both expression levels of all proteins and the growth rate start to fluctuate until the cell finds a state of protein expression that realizes a high growth rate. In the simulation, the noise amplitude σ = 0.2.
Figure 5.
The Relationship between the Noise Amplitude σ and the Growth Rate vg
Starting from randomly chosen initial conditions against the noise amplitude σ ranging 10−4 < σ <3, the growth rates vg after 105 time steps are plotted. In the intermediate range σ of the noise strength 10−2 < σ <1, cellular states with high growth rates are selected among a huge number of possible cellular states, as depicted in Figure 2.