Figure 1.
Two enzymes, Ea and Ed, covalently (de)modify the components X and X*, respectively. The activating enzyme Ea provides the input signal, the unmodified component X is the detection component, and the modified component X* provides the output signal.
Figure 2.
The Effect of Enzyme Localization on the Response of a Push–Pull Network
The input–output relation of the push–pull network shown in Figure 1 is plotted for different values of the total substrate concentration [S]T, for the case in which all components are uniformly distributed in space (A) and for the case in which the activating enzyme is located at one end of the cell, while the other components can diffuse freely through the cell (B). Here, . In (A) and (B), [Ed]T = 0.5 μM, KM,a = KM,d = 0.5 μM, and k3 = k6 = 25s−1. In (B), the diffusion constant is D = 10 μm2 s−1. The inset shows the logarithmic gain g ≡ ∂ln[X*]T / ∂ln[Ea]T. It is seen that the sharpness of the response increases markedly with increasing substrate concentration when all the components are uniformly distributed in space (A), but much less so when the activating enzyme Ea is located at one pole of the cell, while the deactivating enzyme Ea is distributed in the cytoplasm. When both enzymes Ea and Ed are located at one pole, the steady-state dose–response curve is identical to that in (A).
Figure 3.
Effect of the Diffusion Coefficient on the Response
The input–output relation of a network in which the activating enzyme is located at one pole, while the other components can freely diffuse in the cytoplasm, is plotted for different values of the diffusion constant D (in μm2s−1) of the cytoplasmic components. The inset shows the logarithmic gain g ≡ ∂ln[X*]T / ∂ln[Ea]T. It is seen that the gain of the push–pull network strongly increases with increasing diffusion constant. If D → ∞, the dose–response curve approaches that of the push–pull network in which the components are uniformly distributed in space (and that of the network in which the enzymes are colocalized). The total substrate concentration is [S]T = 20 μM, the total concentration of the deactivating enzyme is [Ed]T = 0.5 μM, the Michaelis-Menten constants are KM,a = KM,d = 0.5 μM, and the catalytic rate constants are k3 = k6 = 25s−1.
Figure 4.
Concentration Profiles of a Spatially Non-Uniform Push–Pull Network
The concentration profiles of X* (A) and EdX* (B) in a push–pull network in which the activating enzyme is located at one pole of the cell, while the other components are distributed in the cytoplasm, for three different concentrations of the activating enzyme. For all curves, [S]T = 20 μM, [Ed]T = 0.5 μM, KM,a = KM,d = 0.5 μM, k3 = k6 = 25s−1, and D = 10 μm2 s−1.
Figure 5.
Effect of Diffusion on the Concentration Profiles: Weak versus Strong Activation
Profiles of [X*] (A,D), [EdX*] (B,E), and [Ed] (C,F).
(A–C) Low concentration of activating enzyme, [Ea]T = 0.5[Ed]T.
(D–F) High activating enzyme concentration, [Ea]T = 1.5[Ed]T.
For the other parameter values, see Figure 4.
Figure 6.
Response Curves at Different Positions in the Cell
Dose–response curves of the push–pull network in which the activating enzyme is localized at one pole of the cell, while the other components diffuse in the cytoplasm, for different positions in the cell (x = 0 corresponds to the black left most curve, while x = 3 μm corresponds to the black right most curve). Profiles of [X*] (A) and profiles of [EdX*] (B); note that the response becomes sharper farther away from the pole. The green curves correspond to the average or integrated response of the non-uniform system, while the red curves correspond to the uniform system. The inset shows the logarithmic gain g ≡ ∂ln[X*] / ∂ln[Ea]T at the respective positions in the cell (x = 0,1,2,3, μm). For the parameter values, see Figure 4.