Nonequilibrium Thermodynamics and Nonlinear Kinetics in a Cellular Signaling Switch

We develop a rigorous nonequilibrium thermodynamics for an open system of nonlinear biochemical reactions responsible for cell signal processing. We show that the quality of the biological switch consisting of a phosphorylation-dephosphorylation cycle, such as those in protein kinase cascade, is controlled by the available intracellular free energy from the adenosine triphosphate (ATP) hydrolysis in vivo: $\Delta G=k_{B}T\ln ([\mathrm{A}\mathrm{T}\mathrm{P}]/K_{\mathrm{e}\mathrm{q}}[\mathrm{A}\mathrm{D}\mathrm{P}])$, where $K_{\mathrm{e}\mathrm{q}}$ is the equilibrium constant. The model reveals the correlation between the performance of the switch and the level of $\Delta G$. The result demonstrates the importance of nonequilibrium thermodynamics in analyzing biological information processing, provides its energetic cost, establishes an interplay between signal transduction and energy metabolism in cells, and suggests a biological function for phosphoenergetics in the ubiquitous phosphorylation signaling.

Figure 1

The kinetic scheme for a typical PdPC in cell signaling. $A$ is a signaling molecule in its inactive form; $A^{*}$ is the phosphorylated, active form. Several pathways exist between $A$ and $A^{*}$. Pathway I involves a biologically specific kinase for the phosphorylation, i.e., $A+\mathrm{A}\mathrm{T}\mathrm{P}\rightleftharpoons A^{*}+\mathrm{A}\mathrm{D}\mathrm{P}$; pathway II is the dephosphorylation due to a specific phosphatase: $A^{*}\rightleftharpoons A+\mathrm{P}\mathrm{i}$. Hence II is not the reverse reaction of I; they are different biochemical reactions. Pathway 0 represents all the background, nonspecific kinases, and phosphatases. As a signaling module, the kinase acts as a stimulus ($S$) and the phosphatase acts as an inhibitor ($I$) for the PdPC. The dashed line represents a positive feedback from $A^{*}$ to the kinase activity [7].

Figure 2

The steady state of the kinetics in Fig. 1 as a function of phosphorylation potential $\Delta G=k_{B}T\ln \gamma$, stimulus strength $[S]$, and inhibitor strength $[I]$. All computations are done with the parameter set $a=10$, which sets the scale for the unit for all concentrations. $\sigma =0.01$, $k_{-1}=10$, ${\tilde{k}}_{-2}=0.1$, $k_{-3}=1$. (a),(b) $[S]=10$. (c) $[I]=1.0$. (a) For $\gamma =1$, no matter what are the positive $[S]$ and $[I]$, there is only one stable steady state: $[A^{*}{]}_{\mathrm{s}\mathrm{s}}=\sigma a/(1+\sigma )$ as expected for a simple chemical equilibrium. The unphysical negative $[I]$ region is shown to illustrate that the closed system $(\gamma =1)$ is structurally unstable mathematically. When $\gamma >1$, multiple steady states appear. For example, in (b) when $[S]=10$ and $\gamma >100$, there are either one steady state, or three steady states. (d) shows the parameter region in which there are three steady states, which represents the hysteresis.

Figure 3

The fault safety factor as a function of the available cellular energy $\Delta G$, computed under the same condition as in Fig. 2. The minimal $\Delta G$ for the switching behavior under this set of conditions is $2\mathrm{k}\mathrm{c}\mathrm{a}\mathrm{l}/\mathrm{m}\mathrm{o}\mathrm{l}$. The fault safety factor, defined as the fractional fluctuations in $[S]$ or $[I]$ the system can tolerate when near the threshold, increases with the energy supply.