Allovalency revisited: an analysis of multisite phosphorylation and substrate rebinding

One salient example comes from the regulation of the cell cycle by ubiquitin mediated protein degradation, a key motif in the control of the cell cycle 5,6 . In the seminal work by Nash et al. 7 , the authors show that the CDK inhibitor, Sic1 functions through a thresholding mechanism -Sic1 must be phosphorylated at least 6 six (of its 9 possible) sites in order to be ubiquitinated and subsequently targeted for degradation.

Sic1 is intrinsically disordered 8 and the location and specificity of these six phosphorylation sites seems to be unimportant at least to some extent. This observation among others 9 led to the hypothesis that the function of these seemingly redundant post translational modifications may be to increase the probability that Sic1 rebinds to its substrate upon disassociation 10,11 and a mathematical model 10 was developed to investigate the rebinding of a polyvalent ligand. In this model, a ligand, once disassociated, effectively escapes from its receptor unless it is phosphorylated a sufficient number of times so as to increase its chances of rebinding.

The problem of ligand rebinding has been extensively studied in many contexts 12- 17 . Some of the most comprehensive studies were carried out in the context of two settings: 1.) ligand binding/unbinding to and from a surface as a model for the kinetics of ligand binding to cell-surface receptors 13,15,18 and 2.) chemotaxis and autocrine signaling resulting in rebinding of a ligand secreted from a cell 12,19,20 . In each of these studies, it was demonstrated that ligand rebinding can be very significant. Despite these advances, how changes in the phosphorylation state of a substrate is related to rebinding and how this affects a biological dose response curve has not been fully investigated. A schematic of this effect is shown in Fig. 1.

Towards this end, we use an integral equation theory and Monte Carlo simulations to study the rebinding of a ligand to a receptor from which it initially disassociated and how this rebinding may be affected by multiple recognition sites. From considering only the effects of a single molecule rebinding to its receptor, we compute the time dependence of the probability that a ligand remains bound as a function of the number of phosphorylations. In turn, we compute the probability that a ligand escapes its target as function of the number of recognition sites. The model and numerical simulations predict that this escape probability can decrease nearly exponentially as a function of the number of independent binding sites thus suggesting that ligand rebinding greatly affects the binding kinetics. We also highlight the importance of two physical regimes of ligand rebinding that are characterized by weak and strong rebinding and show how each regime may affect the input-output relationships of a system with multiple phosphorylation sites. We further note that the model predicts that, although a ligand’s propensity to immediately rebind, as a function of the number of available binding sites, greatly affects the shape of the biological response, additional mechanistic ingredients appear to be required to achieve a highly cooperative response. Finally, we note that while our model predicts that the probability of a polyvalent ligand escaping from its receptor decreases exponentially as a function of the number of binding sites, this property appears insufficient to give rise to a highly cooperative response as has been previous predicted 10 . The source of this discrepancy appears to lie in how the rate constants in the previous phenomenological model were varied independently to achieve the desired cooperativity.

The key considerations that are used to develop our model lie in the questions that we wish to address in this study. In particular, our aim is to investigate how ligand rebinding may be affected by multisite phosphorylation. Other studies of multisite phosphorylation have investigated the consequences of other physical effects such as distributive phosphorylation and feedback regulation 4,21 . We are interested in computing the probability that a ligand remains bound as a function of time and as a function of the number of recognition sites on the receptor.

To model this scenario, we assume that at time zero, a ligand is bound to its receptor and can be released with a constant unit time probability. When the ligand is in immediate proximity

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