Barrier height, entropy production rate and phase coherence at different diffusion coefficient <i>D</i>.

<p>(A), (B), (C) show barrier height, entropy production rate and coherence versus diffusion coefficient <i>D</i> separately.</p

Parameter values of predator prey model.

<p><i>DD</i> is a dilution rate and calculated with the relation where F is a fraction of dilution and <i>T</i> is the time between each dilution event <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0017888#pone.0017888-Balagadde2" target="_blank">[27]</a>.</p

Landscape and probabilistic flux for oscillation state.

<p>(A) shows the 2-dimensional landscape and probabilistic flux for oscillation state at D = 0.001, IPTG = 5, DD = 0.1125. Magenta arrows represent the flux flow vector, green arrows represent the negative gradient of potential energy. (B) shows the 3-dimensional landscape for predator prey network.</p

Landscape for monostable state.

<p>(A) shows 3-dimensional Landscape for monostable state at D = 0.001, IPTG = 5, DD = 0.02 using variables <i>Ae</i>1 and <i>Ae</i>2. (B) shows RR (robustness ratio) versus external noise D for monostable state using the same parameters values.</p

Barrier height, entropy production rate and phase coherence at different perturbation of parameters.

<p>(A), (B), (C) show barrier height, entropy production rate and coherence versus perturbation level(<i>lp</i>) separately.</p

The definition of phase coherence.

<p>The definition of phase coherence.</p

Landscape Topography Determines Global Stability and Robustness of a Metabolic Network

Metabolic networks have gained broad attention in recent years as a result of their important roles in biological systems. However, how to quantify the global stability of the metabolic networks is still challenging. We develop a probabilistic landscape approach to investigate the global natures of the metabolic system under external fluctuations. As an example, we choose a model of the carbohydrate metabolism and the anaplerotic synthesis of oxalacetate in <i>Aspergillus niger</i> under conditions of citric acid accumulation to explore landscape topography. The landscape has a funnel shape, which guarantees the robustness of system under fluctuations and perturbations. Robustness ratio (<i>RR</i>), defined as the ratio of gap between lowest potential and average potential versus roughness measured by the dispersion or square root of variations of potentials, can be used to quantitatively evaluate the global stability of metabolic networks, and the larger the <i>RR</i> value, the more stable the system. Results of the entropy production rate imply that nature might evolve such that the network is robust against perturbations from environment or network wirings and performs specific biological functions with less dissipation cost. We also carried out a sensitivity analysis of parameters and uncovered some key network structure factors such as kinetic rates or wirings connecting the protein species nodes, which influence the global natures of the system. We found there is a strong correlation between the landscape topography and the input-output response. The more stable and robust the metabolic network is, the sharper the response is

The phase plane portrait for the predator-prey network in terms of parameter <i>IPTG</i> and <i>DD</i>.

<p>The phase plane portrait for the predator-prey network in terms of parameter <i>IPTG</i> and <i>DD</i>.</p

Sensitivity analysis.

<p>(A) shows the effects of parameters on the barrier height at the same perturbation. x axis represent: 1:kc1, 2:kc2, 3:dc1, 4:dc2, 5:K1, 6:K2, 7:kA1, 8:kA2, 9:dAe1, 10:dAe2. (B) shows respectively the effect of 6 parameters on barrier height. Δ<i>k/k</i> represents the percent of parameters increased.</p

Distribution of amplitude and period.

<p>(A), (B) show the distribution of amplitude and period at different diffusion coefficient <i>D</i> separately.</p