arXiv subjects
Publications and source records attributed to Pranay Talla.
In this short note, we prove that $\alpha$-concavity of the pressure is not preserved for the porous medium equation in dimensions $n=3$ and higher for any $\alpha\in [0,1]\backslash \{\frac{1}{2}\}$. Together with the result of Chau-Weinkove for $n=2$, this fully resolves an open problem posed by V\'asquez on whether pressure concavity is preserved in general for the porous medium equation.
Switch-like motifs are among the basic building blocks of biochemical networks. A common motif that can serve as an ultrasensitive switch consists of two enzymes acting antagonistically on a substrate, one making and the other removing a covalent modification. To work as a switch, such covalent modification cycles must be held out of thermodynamic equilibrium by continuous expenditure of energy. Here, we exploit the linear framework for timescale separation to establish tight bounds on the performance of any covalent-modification switch, in terms of the chemical potential difference driving the cycle. The bounds apply to arbitrary enzyme mechanisms, not just Michaelis-Menten, with arbitrary rate constants, and thereby reflect fundamental physical constraints on covalent switching.