Box 1 Modeling Ultrasensitivity with Stimulus Response Curves

The shape of a given systems stimulus-response curve is a key aspect of the steady-state behavior of a signaling system. Typical Michaelis-Menten enzymes exhibit hyperbolic stimulus response curves (Fig. 3, nH = 1). At very low stimulus levels, the response grows linearly with the stimulus. As the extent of the stimulus increases, the response to each quanta of stimulus becomes progressively smaller. In other words, a Michaelian (also referred to as graded or hyperbolic) system obeys the law of diminishing returns. A system that obeys Michaelian sensitivity requires an 81-fold increase in input stimulus to drive it from 10% to 90% maximal activation. By contrast, some systems can achieve sigmoidal stimulus-response curves, and this can be well approximated by the Hill equation:

where nH is the Hill coefficient. In such a system, the first increments of stimulus produce little response, but once the system does begin to respond, it reaches its maximal response rapidly. The higher the Hill coefficient, the more switchlike the response becomes. This is represented graphically in Fig. 3 for Hill coefficients (nH) of 1, 6, and 40.

Biologically, this has several significant outcomes. By incorporating an ultrasensitive biological switch into a signal transduction pathway, a cell can filter out noise in a system as small amounts of stimuli will fail to yield any consequential response. Such a switch could also be used to allow precise control over key decisions in which an exact degree of stimuli rapidly creates a complete response. It also sets a threshold for the signal, allowing the system to effectively collect inputs and precisely monitor when these exceed the threshold level set to convert the system to a complete response. In this last case, the signals are integrated at the level of the switch.

Figure 3 Multiple, independent phosphorylation events of Sic1 by a cyclin-dependent kinase (CDK) create the basis for an ultrasensitive biological switch for the onset of DNA replication in yeast. Phosphorylation of the CDK inhibitor Sic1by the Cln1/2-Cdc28 kinase on at least six independent sites is required for the productive interaction of Sic1 with the WD40 repeat region of the Cdc4 F-box protein. Binding of Sic1 to Cdc4 allows the ubiquitination of Sic1 by the SCF E3 ubiq-uitin protein ligase complex and subsequent degradation of Sic1. The requirement for multisite phosphorylation results in a sigmoidal stimulus-response curve with a Hill coefficient (nH) of six, forming the basis for an ultrasensitive biological switch. Additional factors in the biological milieu of the cell likely conspire to significantly improve the degree of ultrasensitivity, as indicated by the curve shown with a Hill coefficient of 40. By contrast, a single phosphorylation event, or simple protein-protein interaction is inherently Michaelian in nature with nH = 1.

Figure 3 Multiple, independent phosphorylation events of Sic1 by a cyclin-dependent kinase (CDK) create the basis for an ultrasensitive biological switch for the onset of DNA replication in yeast. Phosphorylation of the CDK inhibitor Sic1by the Cln1/2-Cdc28 kinase on at least six independent sites is required for the productive interaction of Sic1 with the WD40 repeat region of the Cdc4 F-box protein. Binding of Sic1 to Cdc4 allows the ubiquitination of Sic1 by the SCF E3 ubiq-uitin protein ligase complex and subsequent degradation of Sic1. The requirement for multisite phosphorylation results in a sigmoidal stimulus-response curve with a Hill coefficient (nH) of six, forming the basis for an ultrasensitive biological switch. Additional factors in the biological milieu of the cell likely conspire to significantly improve the degree of ultrasensitivity, as indicated by the curve shown with a Hill coefficient of 40. By contrast, a single phosphorylation event, or simple protein-protein interaction is inherently Michaelian in nature with nH = 1.

recognized by the phospho-dependent interaction domain of an E3 protein-ubiquitin ligase and is thereby recruited into a ubiquitination complex [58].

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