The probability flux flowing in the 2¹⁰ states space of the fission yeast cell cycle.

<p>The thickness of the arrows is proportional to the magnitude of the probability fluxes and the node size is related to the steady state probability of the state. The colored flux loops represent several typical flux loops among all, in which the largest blue loop with 10 nodes stands out and becomes the “native” biological cycle. The simulations were performed with <i>c</i> = 0.001, <i>μ</i> = 5, <i>γ</i> = 60%.</p

Influence on the system robustness from the variation of the perturbation parameter <i>c</i>, by fixing <i>μ</i> = 5, <i>γ</i> = 60%.

<p>(a) Steady-state probability of “native” cycle (<i>P</i>_{<i>Circle</i>}) versus <i>c</i>. (b) Robustness Ratio (RR) versus <i>c</i>. (c) Entropy production rate (<i>dS</i>/<i>dt</i>) versus <i>c</i>. (d) Entropy production rate (<i>dS</i>/<i>dt</i>) versus Robustness Ratio.</p

Influence on the probability flux by changing the cycling activation strength <i>γ</i> which represents the jumping probability from the G1/G0 state to the activated G1 state (START phase), while fixing <i>μ</i> = 0.8, <i>c</i> = 0.001.

<p>(a) Steady-state probability flux versus <i>γ</i>. (b) Robustness Ratio (RR) of flux spectrum versus <i>γ</i>. (c) Entropy production rate (<i>dS</i>/<i>dt</i>) versus RR of flux. (d) Robustness Ratio (RR) of flux spectrum versus steady-state probability of “native” cycle (<i>P</i>_{<i>Circle</i>}).</p

Transition probability <i>T</i>(<i>s</i>_<i>i</i>(<i>t</i>′)|<i>S</i>(<i>t</i>)) of gene node <i>i</i>.

<p>Transition probability <i>T</i>(<i>s</i>_<i>i</i>(<i>t</i>′)|<i>S</i>(<i>t</i>)) of gene node <i>i</i>.</p

The minimal robustness backbone subnetworks of the fission yeast networks.

<p>All the links marked by red color contribute a backbone subnetwork, which is generated from the evaluation of robustness in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005710#pcbi.1005710.t003" target="_blank">Table 3</a> (marked by italic type). The remaining links form a residual auxiliary subnetwork. All the signs of the links are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005710#pcbi.1005710.g001" target="_blank">Fig 1</a>.</p

Probability flux spectrum of all the loops, where <i>μ</i> = 5, <i>c</i> = 0.001 and <i>γ</i> = 60%.

<p>The flux of the loop which is formed by the 10 states of the biological pathway and thus represents the “native” cycle is drawn in green line. It is the lowest one in negative logarithm compared to other cycle loops.</p

Steady-state probability of stationary G1 (<i>P</i>_<i>G</i>1) (a) and probability of “native” cycle (<i>P</i>_{<i>Circle</i>}) (b) versus the variation of <i>μ</i> under different jumping probabilities <i>γ</i> by fixing <i>c</i> = 0.001.

<p>Steady-state probability of stationary G1 (<i>P</i>_<i>G</i>1) (a) and probability of “native” cycle (<i>P</i>_{<i>Circle</i>}) (b) versus the variation of <i>μ</i> under different jumping probabilities <i>γ</i> by fixing <i>c</i> = 0.001.</p

The cell cycle regulatory network of fission yeast with 10 gene nodes.

<p>The solid arrow (→) represents positive regulations between gene nodes. The inhibition sign (⋅⋅⋅|)represents negative regulations between gene nodes.</p

Influence on the system robustness from the variation of the sharpness of the response or the inverse noise level <i>μ</i>, by fixing <i>c</i> = 0.001 and <i>γ</i> = 60%.

<p>(a) Steady-state probability of stationary G1 (<i>P</i>_<i>G</i>1) and “native” cycle (<i>P</i>_{<i>Circle</i>}) versus <i>μ</i>. (b) Robustness Ratio (RR) versus <i>μ</i>. (c) Entropy production rate (<i>dS</i>/<i>dt</i>) versus <i>μ</i>. (d) Entropy production rate (<i>dS</i>/<i>dt</i>) versus Robustness Ratio.</p

Three dimensional potential landscape and two dimensional contour in projected 2 dimensional state space.

<p>The vertical axis and color represent the potential level of each state in the three dimension and the contour map laying on the bottom respectively. The low potential valley of the potential is a cycle or closed ring, which is exactly the biological cycle path with low potential level, and this can also be seen more clearly on the contour map.</p