Protein Scaffolds Can Enhance the Bistability of Multisite Phosphorylation Systems

The phosphorylation of a substrate at multiple sites is a common protein modification that can give rise to important structural and electrostatic changes. Scaffold proteins can enhance protein phosphorylation by facilitating an interaction between a protein kinase enzyme and its target substrate. In this work we consider a simple mathematical model of a scaffold protein and show that under specific conditions, the presence of the scaffold can substantially raise the likelihood that the resulting system will exhibit bistable behavior. This phenomenon is especially pronounced when the enzymatic reactions have sufficiently large KM, compared to the concentration of the target substrate. We also find for a closely related model that bistable systems tend to have a specific kinetic conformation. Using deficiency theory and other methods, we provide a number of necessary conditions for bistability, such as the presence of multiple phosphorylation sites and the dependence of the scaffold binding/unbinding rates on the number of phosphorylated sites. Author Summary: The modification of a protein at multiple sites can result in a number of interesting behaviors at the cellular level, such as all-or-none responses to an external input, or two different stable cellular states in otherwise identical environments. Such behaviors can aid in many different forms of cellular decision-making, e.g., cell differentiation or cell division. In this paper, we show that bistable behavior can be greatly enhanced by the presence of a scaffold protein, which binds to the substrate protein and either relocates it or otherwise affects the action of the modifying enzymes. The scaffold protein substantially widens the range of parameters for which bistability is observed when, a key descriptor of enzymatic activity, assumes medium to large values found in a majority of enzymes. Indeed, when was greater than the concentration of the target substrate, bistability was never observed in the absence of a scaffold. In addition to extensive computational work, we also carried out a mathematical analysis of a simplified system in order to identify the conditions under which bistability is possible. We conclude that scaffold proteins can be a simple yet very useful addition to multisite protein systems when bistability is advantageous.