Energy dissipation drives the gradient signal amplification through an incoherent type-1 feed-forward loop

We present here the analytical relation between the gain of eukaryotic gradient sensing network and the associated thermodynamic cost. By analyzing a general incoherent type-1 feed-forward loop, we derive the gain function (G) through the reaction network and explicitly show that G depends on the nonequilibrium factor (0≤γ≤1 with γ=0 and 1 representing irreversible and equilibrium reaction systems, respectively), the Michaelis constant (K_{M}), and the turnover ratio (r_{cat}) of the participating enzymes. We further find the maximum possible gain is intrinsically determined by K_{M}/G_{max}=(1/K_{M}+2)/4. Our model also indicates that the dissipated energy (measured by -lnγ), from the intracellular energy-bearing bioparticles (e.g., ATP), is used to generate a force field F_{γ}∝(1-sqrt[γ]) that reshapes and disables the effective potential around the zero gain region, which leads to the ultrasensitive response to external chemical gradients.