Compact modeling of allosteric multisite proteins: application to a cell size checkpoint

Bounds on the Ultrasensitivity of Biochemical Reaction Cascades

The ultrasensitivity of a dose response function can be quantifiably defined using the generalized Hill coefficient of the function. We examined an upper bound for the Hill coefficient of the composition of two functions, namely the product of their individual Hill coefficients. We proved that this upper bound holds for compositions of Hill functions, and that there are instances of counterexamples that exist for more general sigmoidal functions. Additionally, we tested computationally other types of sigmoidal functions, such as the logistic and inverse trigonometric functions, and we provided computational evidence that in these cases the inequality also holds. We show that in large generality there is a limit to how ultrasensitive the composition of two functions can be, which has applications to understanding signaling cascades in biochemical reactions.

Effect of magnitude and variability of energy of activation in multisite ultrasensitive biochemical processes

Protein activity is often regulated by ligand binding or by post-translational modifications such as phosphorylation. Moreover, proteins that are regulated in this way often contain multiple ligand binding sites or modification sites, which can operate to create an ultrasensitive dose response. Here, we consider the contribution of the individual modification/binding sites to the activation process, and how their individual values affect the ultrasensitive behavior of the overall system. We use a generalized Monod-Wyman-Changeux (MWC) model that allows for variable conformational free energy contributions from distinct sites, and associate a so-called activation parameter to each site. Our analysis shows that the ultrasensitivity generally increases as the conformational free energy contribution from one or more sites is strengthened. Furthermore, ultrasensitivity depends on the mean of the activation parameters and not on their variability. In some cases, we find that the best way to maximize ultrasensitivity is to make the contribution from all sites as strong as possible. These results provide insights into the performance objectives of multiple modification/binding sites and thus help gain a greater understanding of signaling and its role in diseases.

A conserved signaling network monitors delivery of sphingolipids to the plasma membrane in budding yeast

In budding yeast, signals generated in response to membrane growth are required for cell cycle progression. A mass spectrometry screen for signals triggered by an arrest of membrane growth identified sphingolipid signaling pathways. Delivery of sphingolipids to the plasma membrane could generate signals that control cell growth and the cell cycle.

In budding yeast, cell cycle progression and ribosome biogenesis are dependent on plasma membrane growth, which ensures that events of cell growth are coordinated with each other and with the cell cycle. However, the signals that link the cell cycle and ribosome biogenesis to membrane growth are poorly understood. Here we used proteome-wide mass spectrometry to systematically discover signals associated with membrane growth. The results suggest that membrane trafficking events required for membrane growth generate sphingolipid-dependent signals. A conserved signaling network appears to play an essential role in signaling by responding to delivery of sphingolipids to the plasma membrane. In addition, sphingolipid-dependent signals control phosphorylation of protein kinase C (Pkc1), which plays an essential role in the pathways that link the cell cycle and ribosome biogenesis to membrane growth. Together these discoveries provide new clues as to how growth-­dependent signals control cell growth and the cell cycle.

N-Site phosphorylation systems with 2n-1 steady states

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of n-site sequential distributive phosphorylation are therefore studied frequently. In particular, in Wang and Sontag (J Math Biol 57:29–52, 2008), it is shown that models of n-site sequential distributive phosphorylation admit at most 2n - 1 steady states.Wang and Sontag furthermore conjecture that for odd n, there are at most n and that, for even n, there are at most n + 1 steady states. This, however, is not true: building on earlier work in Holstein et al. (BullMath Biol 75(11):2028–2058, 2013), we present a scalar determining equation for multistationarity which will lead to parameter values where a 3-site system has 5 steady states and parameter values where a 4-site system has 7 steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag.We furthermore study the inherent geometric properties of multistationarity in n-site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein.