Self-regulation in a minimal model of chemical self-replication

Abstract

In biological systems, regulation plays an important role in keeping metabolite concentrations within physiological ranges. To study the dynamical implications of self-regulation, we consider a functional form used in genetic networks and couple it to a mechanism associated with chemical self-replication. For the two-variable minimal model, we find that activation can yield chemical toggles similar to those reported for gene repression in E. coli as well as more complex dynamics.

Keywords: Bistability; Chemical self-replication; Genetic regulation.

Fig. 1

Regulated source, R(b) with r₁ = 0, α₁ = 0.20, K_r = 0.20, and β = 1 for the pure repression case, (m = 0, n = 2), and the activation with a saturation case (2,2)

Fig. 2

Steady-state solutions for pure repression as a function of k₂, with m = 0,n = 2, r = 0.10, k_u = 0.01, and β = 1. aK₂ = 0.30 and K = 0.10; bK₂ = 0.05, and K = 0.20

Fig. 3

Steady-state solutions for activation with saturation as a function of k₂ with m = n = 2, r = 0.10, K₂ = 0.05, K = 0.20, k_u = 0.01, and β = 1. The figure on the right is an enlargement of the left-hand figure, showing the three-root region of (29)

Fig. 4

Bifurcation diagrams for k₂ = 0 and m = 0, n = 2, K₂ = 0.05, k_u = 0.01, and β = 1. Thick dark line encloses the region of oscillations, thin dark line shows stable steady states, and thin light line, inside the oscillatory region, shows unstable steady states. a One parameter, r, diagram with K = 0.20, and b Two parameter diagram, (r,K)

Fig. 5

Bifurcation diagrams for k₂ = 0.10 and m = 0, n = 2, K₂ = 0.05, and k_u = 0.01. For meaning of the line types, see Fig. 4. a One parameter, r, diagram with K = 0.20, and b two parameter diagram, (r,K)

Fig. 6

Bistability for m = n = 2, r = 0.10, k₂ = 0.80, K₂ = 0.05, K = 0.20, k_u = 0.01, and β = 1. The blue dots show the three steady states and the black lines show the trajectories of four different initial conditions. Two of the initial conditions are close to the separatrix

Fig. 7

Bifurcation diagram for activation with saturation and leak transcription, where r is the bifurcation parameter with m = n = 2, K₂ = 0.10, K = 0.20, k_u = 0.01, and β = 1. ak₂ = 0.20, and bk₂ = 0.40. For interpretation of the lines, see Fig. 4

Fig. 8

Bifurcation diagram for activation with saturation and leak transcription, where r is the bifurcation parameter with m = n = 2, K₂ = 0.10, K = 0.20, k_u = 0.01, and β = 1. ak₂ = 0.50, and bk₂ = 0.60. For interpretation of the lines, see Fig. 4

Fig. 9

Bifurcation diagram for activation with saturation and leak transcription, where r is the bifurcation parameter. am = n = 2, K₂ = 0.10, K = 0.20, β = 1, k_u = 0.01, and k₂ = 0.50, and b amplification of the figure near the saddle-node. For interpretation of the lines, see Fig. 4

Fig. 10

Bifurcation diagram for activation with saturation and leak transcription, where r is the bifurcation parameter. am = n = 2, K₂ = 0.05, K = 0.20, β = 1, k_u = 0.01, and k₂ = 0.50. b Amplification of the figure near the saddle-node. For interpretation of the lines, see Fig. 4. Notice the region of coexistence between stable limit cycles and stable steady states, which could be a source of hysteresis

Fig. 11

a Two-parameter bifurcation diagram for activation with saturation and leak transcription, where r and k₂ are the bifurcation parameters. Other parameters are: m = n = 2, K₂ = 0.05, K = 0.20, β = 1, and k_u = 0.01. b Amplification of the figure near the saddle node (SN) bifurcations showing the PAH bifurcation