Multisite protein phosphorylation makes a good threshold but can be a poor switch

Abstract

Phosphorylation and dephosphorylation play a fundamental role in eukaryotic signaling. Some 30% of proteins are phosphorylated at any time, many on multiple sites, raising the question of how the cellular phosphorylation state is regulated. Previous work for one and two phosphorylation sites has revealed mechanisms, such as distributive phosphorylation, for switch-like regulation of maximally phosphorylated phosphoforms. These insights have led to the influential view that more phosphorylation sites leads to steeper switching, as proposed for substrates like cyclin E and the cyclin-dependent kinase inhibitor Sic1. An analytical study of the ordered distributive case reveals a more complex story. Multisite phosphorylation creates an efficient threshold: The proportion of maximally phosphorylated substrate is maintained close to 0 when the ratio of kinase to phosphatase activity lies below a suitable threshold, and this threshold increases with increasing numbers of sites, n. However, above the threshold, the response may not always abruptly switch between 0 and 1, as would be the case for an efficient switch, but may increase in a gradual manner, which becomes more hyperbolic with increasing n. Abrupt switching cannot be attributed merely to n being large. We point out that conventional measures of ultrasensitivity must be modified to discriminate between thresholding and switching; we discuss additional factors that influence switching efficiency and suggest new directions for experimental investigation.

Fig. 1.

Hill curves h_n(u) = uⁿ/(1 + uⁿ), for n = 1, 2, and 10, illustrating the current view of distributive phosphorylation. The horizontal axis represents some unspecified measure of kinase activity relative to phosphatase activity. The bold curve is the limit as n → ∞: the “perfect switch.”

Fig. 2.

Behavior of ordered distributive P&D with n sites, showing ρ_n(u; 1), as defined in Eq. 10, for n = 1, 2, and 10. The bold curve is ρ(u; 1), defined in Eq. 11 as the limit of ρ_n(u; 1) as n → ∞.

Fig. 3.

The Hill curves, h_n(u) = uⁿ/(1 + uⁿ) (solid line), compared with the multisite phosphorylation curves, ρ_n(u; δ,..., δ, δ^1-n), defined in Eq. 13 (dashed line), for δ = 0.1 and n = 2, 5, and 10.