One third of all proteins in eukaryotic cells are phosphorylated at any time 
. Phosphorylation profile has been interpreted as a “molecular barcode” 
to direct protein for other processes such as activation, inactivation, translocation, and degradation. In particular, three major transitions in the cell cycle, namely, entry into the S phase, separation of sister chromatids, and exit from mitosis, involve degradation of certain proteins after phosphorylation-dependent ubiquitination 
. Phosphorylation-driven ubiquitination through the SCF pathway and subsequent proteasomal degradation have been considered as a biochemical switch crucial for coordinating phase changes during the cell cycle 
. For instance, the ubiquitination and degradation of Sic1, a cyclin-dependent kinase inhibitor in yeast, and its functional homolog in mammalian cells, p27, are triggered by phosphorylation 
. Degradation of Sic1 and p27 leads to the release of cyclins required for DNA synthesis in the S phase. Reduced nuclear p27 is observed in up to 60% of primary human breast cancers, which has been correlated with increased activity of the Src kinase family 
The number of phosphorylation sites observed in one protein can vary from 1 to over 100 
. It has become increasingly apparent that multisite phosphorylation is a widespread phenomenon among regulatory proteins in eukaryotic cells. Multisite protein phosphorylation potentially provides a precise tool for dynamic regulation of the downstream process. Different phosphorylation profiles of a single protein might be linked to different functions. For example, the retinoblastoma protein (Rb) has 16 Ser/Thr-Pro phosphorylation sites and interacts with different proteins during various cell cycle phases depending upon its phosphorylation profile 
. Most notably, in early G1, Rb is hypophosphorylated and sequesters the E2F family of transcription factors, thereby preventing the transcription of genes required for S-phase entry; while in late G1, Rb becomes hyperphosphorylated and thus inactive in repressing G1/S transition 
. As another example, two sites in Xenopus
protein Wee1A are responsible for the mitotic inactivation of this protein, while at least two others regulate its proteolysis during interphase 
. Intriguingly, as far as protein stability is concerned, an alternative property of multisite phosphorylation, the degree of phosphorylation (i.e. the number of phosphate groups on a protein), instead of the exact pattern, might determine the protein's fate. A well-studied example is protein Sic1, which plays a key role in regulating the G1/S transition in the cell cycle of Saccharomyces cerevisiae
Sic1 inhibits Clb5,6-Cdc28 kinase required for DNA replication and is believed to provide precise timing for the G1 to S transition by undergoing switch-like proteasome-mediated degradation upon phosphorylation by the Cln2-Cdc28 kinase complex. Yeast strains lacking Sic1 initiate DNA replication earlier and show extended S phase 
. On the other hand, in mutant strains that are resistant to Sic1 degradation, cells experience lengthened G1 phase in an otherwise wild-type genetic background 
or G1 phase arrest in more complex situations 
. Sic1 is phosphorylated by the Cln2-Cdc28 kinase complex on nine Ser/Thr-Pro residues 
. Nash et al.
investigated how multisite phosphorylation of Sic 1 regulates its ubiquitination and degradation 
. They began with the Sic1 mutant which lacks all the nine phosphorylation sites and restored the sites one by one in the order of their importance measured by the degree to which elimination of a single site affects the Sic1 turnover. Serial reintroduction of five sites failed to reestablish Sic1 binding to Cdc4, a subunit in the ubiquitin ligase
that determines the target specificity, or cell viability. Astonishingly, re-addition of a sixth seemingly insignificant site abruptly restored Sic1's binding with Cdc4 in vitro
and revived the cells in vivo
. These experiments clearly revealed that there is a threshold number of phosphorylated sites required to render binding of Sic1 with Cdc4. The “counting” mechanism underlying this multisite-dependent digital interaction between phosphorylated Sic1 and Cdc4 has been studied both theoretically and experimentally. Mathematical modeling suggested that cooperative interactions between a disordered multi-phosphorylated Sic1 and a single-site receptor Cdc4 can explain the observed phosphorylation threshold 
. Furthermore, cumulative electrostatic forces resulted from negatively charged phosphate groups were proposed as the physical basis for the digital interaction between Sic1 and Cdc4 
. Recently, NMR analysis showed that Sic1 indeed exists in an intrinsically disordered state and its multiple phosphorylated sites interact with the single receptor site of Cdc4 in dynamic equilibrium 
Upon their remarkable discovery that Sic1 requires at least six sites phosphorylated to bind to Cdc4 for subsequent ubiquitination and degradation, Nash et al.
hypothesized that this phosphorylation threshold eventually causes Sic1 to degrade in a switch-like manner during the G1/S transition 
. Reviewing the above seminal work, Deshaies and Ferrell conjectured more specifically that multisite phosphorylation can create temporal thresholds 
. They calculated time courses for Sic1 destruction in three scenarios: Sic1 destruction triggered by one fast, one slow, or six fast phosphorylations. It was suggested that when six distributive and equivalent phosphorylations are required, Sic1 destruction is initially very slow when the first five sites are getting phosphorylated, then after a lag period, degradation dramatically speeds up. In another word, a temporal threshold is created for Sic1 destruction from the onset of Cln-CDK activation. Alternatively, if the degradation of Sic1 be governed by a single phosphorylation, there would have been a gradual decrease of Sic1 amount without time delay. The modeling framework presented by Deshaies and Ferrell in this review, albeit primitive, represents a very intriguing idea aiming to make the key connection between Sic1's observed phosphorylation threshold number to its ultimate function of regulating the G1/S transition. However, there remained several caveats. First, six phosphorylations were considered, while Sic1 has a total of nine sites. Do the remaining sites play any role? Second, it can be anticipated that exactly how the kinase is activated, i.e. the temporal profile of the kinase signal, affects the degradation of the substrate protein, and this aspect was not discussed. Finally, it was not clear what determined quantitatively the temporal threshold and speed of degradation. In this work, we will attempt to address the above issues, by carrying out systematic and detailed mathematical modeling to examine how multisite phosphorylation might lead to switch-like protein degradation.
Switch-like behaviors have been studied extensively in the steady state domain, where the response of a biological system (e.g. the amount of oxygen bound by the hemoglobin protein in response to the change of oxygen concentration) exhibits the very intriguing property of buffering fluctuations in the stimulus below a threshold and amplifying drastically the change of stimulus above the threshold. This type of switch-like response in the steady state domain has been termed “ultrasensitivity” in the literature and a number of mechanisms have been proposed to account for its sources, including ligand cooperativity 
, multi-step effect 
, enzyme saturation (i.e. zero-order ultrasensitivity) 
, positive feedback 
, multi-level cascade 
, multisite phosphorylation 
, and substrate competition 
. These studies have generated important insights concerning steady-state responses, which often correspond to in vitro
experimental assays, on how switch-like behaviors arise. Nevertheless, what is crucial for many systems, particularly in vivo
processes, is the transient stimulus-response curve such as the temporal profile of Sic1 during the G1/S transition, whereas switch-like responses in the temporal domain have been investigated very limitedly. It should be noted that the cell cycle signaling network in yeast has been examined very extensively through mathematical modeling 
; however, degradation of macromolecules such as Sic1 has been modeled with a single phosphorylation reaction without taking into account the multiple phosphorylation steps.
In a previous work, we showed that multisite phosphorylation is a potential source of switch-like steady-state responses; most importantly, a large number of total sites combined with an intermediate threshold number of sites for changing substrate functionality account for the switch-like behavior (manuscript in revision). Here, extending our previous studies, we investigate switch-like responses in the time domain when protein stability depends on the degree of phosphorylation. We will present a model to analyze phosphorylation-triggered elimination of the substrate protein in response to the rise of kinase activity. We will show quantitatively how different parameters affect the protein elimination dynamics when degradation occurs above a threshold number of phosphorylations. In particular, we will explore how the extent to which the degradation dynamics is switch-like is affected by the type of kinase stimuli, the number of phosphorylation sites, the order of phosphorylation reactions, and kinetic parameters. We have developed the model mainly based on the Sic1 system to reveal the role of its existing multiple sites in regulating the protein's switch-like destruction during the G1/S transition. However, multisite phosphorylation is potentially a widespread source of switch-like protein degradation and the design principles revealed by our model might be applicable to many other multisite regulatory proteins.