Unlimited multistability in multisite phosphorylation systems

Cellular information processing While naked DNA has a relatively static and easy to grasp information capacity — 2 bits per nucleotide—reversible chemical modification at multiple sites in even a single protein encodes a potentially large and so far untractable amount of information. Here Matthew Thomson and Jeremy Gunawardena reduce the 3 × 2 n nonlinear differential equations describing dynamic phosphorylation at n sites on a given protein ( n varying from less than 7 in bacteria to more than 150 in eukaryotes) to just two algebraic equations. The method allows them to estimate the information capacity of a signalling protein as a function of varying amounts of modifying enzymes (kinases and phosphatases). Algebraic geometry could extend the method to diverse and parallel enzymatic modifications such as those governing the 'histone code' of gene regulation. Although naked DNA has a relatively static and easy to grasp information capacity, reversible phosphorylation at several sites in even a single protein encodes a potentially large amount of information, and the calculation of this information capacity is complex. Here, this complexity is reduced to solving two algebraic equations, allowing the estimation of the information capacity of a signalling protein as a function of the varying amounts of kinases and phosphatases. Reversible phosphorylation on serine, threonine and tyrosine is the most widely studied posttranslational modification of proteins 1 , 2 . The number of phosphorylated sites on a protein ( n ) shows a significant increase from prokaryotes, with n ≤ 7 sites, to eukaryotes, with examples having n ≥ 150 sites 3 . Multisite phosphorylation has many roles 4 , 5 and site conservation indicates that increasing numbers of sites cannot be due merely to promiscuous phosphorylation. A substrate with n sites has an exponential number (2 n ) of phospho-forms and individual phospho-forms may have distinct biological effects 6 , 7 . The distribution of these phospho-forms and how this distribution is regulated have remained unknown. Here we show that, when kinase and phosphatase act in opposition on a multisite substrate, the system can exhibit distinct stable phospho-form distributions at steady state and that the maximum number of such distributions increases with n . Whereas some stable distributions are focused on a single phospho-form, others are more diffuse, giving the phospho-proteome the potential to behave as a fluid regulatory network ab