In molecular biology, ultrasensitivity describes an output response that is more sensitive to stimulus change than the hyperbolic Michaelis-Menten response. Ultrasensitivity is one of the biochemical switches in the cell cycle and has been implicated in a number of important cellular events, including exiting G2 cell cycle arrests in Xenopus laevis oocytes, a stage to which the cell or organism would not want to return. Ultrasensitivity is a cellular system which triggers entry into a different cellular state. Ultrasensitivity gives a small response to first input signal, but an increase in the input signal produces higher and higher levels of output. This acts to filter out noise, as small stimuli and threshold concentrations of the stimulus (input signal) is necessary for the trigger which allows the system to get activated quickly. Ultrasensitive responses are represented by sigmoidal graphs, which resemble cooperativity. The quantification of ultrasensitivity is often performed approximately by the Hill equation:
Response = S t i m u l u s n ( EC 50 n + S t i m u l u s n ) {\textstyle {\ce {Response}}={Stimulus^{n} \over ({\ce {EC50}}^{n}+Stimulus^{n})}}
Where Hill's coefficient (n) may represent quantitative measure of ultrasensitive response.
Historical development Zero-order ultrasensitivity was first described by Albert Goldbeter and Daniel Koshland, Jr in 1981 in a paper in the Proceedings of the National Academy of Sciences. They showed using mathematical modeling that modification of enzymes operating outside of first order kinetics required only small changes in the concentration of the effector to produce larger changes in the amount of modified protein. This amplification provided added sensitivity in biological control, and implicated the importance of this in many biological systems. Many biological processes are binary (ON-OFF), such as cell fate decisions, metabolic states, and signaling pathways. Ultrasensitivity is a switch that helps decision-making in such biological processes. For example, in apoptotic process, a model showed that a positive feedback of inhibition of caspase 3 (Casp3) and Casp9 by inhibitors of apoptosis can bring about ultrasensitivity (bistability). This positive feedback cooperates with Casp3-mediated feedback cleavage of Casp9 to generate irreversibility in caspase activation (switch ON), which leads to cell apoptosis. Another model also showed similar but different positive feedback controls in Bcl-2 family proteins in apoptotic process. Recently, Jeyeraman et al. have proposed that the phenomenon of ultrasensitivity may be further subdivided into three sub-regimes, separated by sharp stimulus threshold values: OFF, OFF-ON-OFF, and ON. Based on their model, they proposed that this sub-regime of ultrasensitivity, OFF-ON-OFF, is like a switch-like adaption which can be accomplished by coupling N phosphorylation–dephosphorylation cycles unidirectionally, without any explicit feedback loops. Other recent work has emphasized that not only is the topology of networks important for creating ultrasensitivity responses, but that their composition (enzymes vs. transcription factors) strongly affects whether they will exhibit robust ultrasensitivity. Mathematical modeling suggests for a broad array of network topologies that a combination of enzymes and transcription factors tends to provide more robust ultrasensitivity than that seen in networks composed entirely of transcription factors or composed entirely of enzymes.
Mechanisms Ultrasensitivity can be achieved through several mechanisms:
Multistep mechanisms (examples: cooperativity) and multisite phosphorylation Buffering mechanisms (examples: decoy phosphorylation sites) or stoichiometric inhibitors Changes in localisation (such as translocation across the nuclear envelope) Saturation mechanisms (also known as zero-order ultrasensitivity) Positive feedback Allovalency Non-Zero-Order Ultrasensitivity in Membrane Proteins Dissipative Allostery
Multistep Mechanisms Multipstep ultrasensitivity occurs when a single effector acts on several steps in a cascade. Successive cascade signals can result in higher levels of noise being introduced into the signal that can interfere with the final output. This is especially relevant for large cascades, such as the flagellar regulatory system in which the master regulator signal is transmitted through multiple intermediate regulators before activating transcription. Cascade ultrasensitivity can reduce noise and therefore require less input for activation. Additionally, multiple phosphorylation events are an example of ultrasensitivity. Recent modeling has shown that multiple phosphorylation sites on membrane proteins could serve to locally saturate enzyme activity. Proteins at the membrane are greatly reduced in mobility compared to those in the cytoplasm, this means that a membrane tethered enzyme acting upon a membrane protein will take longer to diffuse away. With the addition of multiple phosphorylation sites upon the membrane substrate, the enzyme can - by a combination of increased local concentration of enzyme and increased substrates - quickly reach saturation.
Buffering Mechanisms Buffering Mechanisms such as molecular titration can generate ultrasensitivity. In vitro, this can be observed for the simple mechanism:
A + B ↽ − − ⇀ AB {\displaystyle {\ce {A + B <=> AB}}}
Where the monomeric form of A is active and it can be inactivated by binding B to form the heterodimer AB. When the concentration of B T {\displaystyle B_{T}} ( = [B] + [AB]) is much greater than the K d {\displaystyle K_{d}} , this system exhibits a threshold determined by the concentration of B T {\displaystyle B_{T}} . At concentrations of A T {\displaystyle A_{T}} ( = [A] +[AB]), lower than B T {\displaystyle B_{T}} , B acts as a buffer to free A and nearly all A will be found as AB. However, at the equivalence point, when A T {\displaystyle A_{T}} ≈ B T {\displaystyle B_{T}} , B T {\displaystyle B_{T}} can no longer buffer the increase in A T {\displaystyle A_{T}} , so a small increase in A T {\displaystyle A_{T}} causes a large increase in A. The strength of the ultrasensitivity of [A] to changes in A T {\displaystyle A_{T}} is determined by B T {\displaystyle B_{T}} / K d {\displaystyle K_{d}} . Ultrasensitivity occurs when this ratio is greater than one and is increased as the ratio increases. Above the equivalence point, A T {\displaystyle A_{T}} and A are again linearly related. In vivo, the synthesis of A and B as well as the degradation of all three components complicates generation of ultrasensitivity. If the synthesis rates of A and B are equal this system still exhibits ultrasensitivity at the equivalence point. One example of a buffering mechanism is protein sequestration, which is a common mechanism found in signalling and regulatory networks. In 2009, Buchler and Cross constructed a synthetic genetic network that was regulated by protein sequestration of a transcriptional activator by a dominant-negative inhibitor. They showed that this system results in a flexibile ultrasensitive response in gene expression. It is flexible in that the degree of ultrasensitivity can be altered by changing expression levels of the dominant-negative inhibitor. Figure 1 in their article illustrates how an active transcription factor can be sequestered by an inhibitor into the inactive complex AB that is unable to bind DNA. This type of mechanism results in an "all-or-none" response, or ultransensitivy, when the concentration of the regulatory protein increases to the point of depleting the inhibitor. Robust buffering against a response exists below this concentration threshold, and when it is reached any small increase in input is amplified into a large change in output.
Changes in localization
Translocation Signal transduction is regulated in various ways and one of the ways is translocation. Regulated translocation generates ultrasensitive response in mainly three ways:
Regulated translocation increases the local concentration of the signaling protein. When concentration of the signaling protein is high enough to partially saturate the enzyme that inactivates it, ultrasensitive response is generated. Translocation of multiple components of the signaling cascade, where stimulus (input signal) causes translocation of both signaling protein and its activator in the same subcellular compartment and thereby generates ultrasensitive response which increases speed and accuracy of the signal. Translocation to the compartment which contains stoichiometric inhibitors. Translocation is one way of regulating signal transduction, and it can generate ultrasensitive switch-like responses or multistep-feedback loop mechanisms. A switch-like response will occur if translocation raises the local concentration of a signaling protein. For example, epidermal growth factor (EGF) receptors can be internalized through clathrin-independent endocytosis (CIE) and/or clathrin-dependent endocytosis (CDE) in ligand concentration-dependent manner. The distribution of receptors into the two pathways was shown to be EGF concentration-dependent. In the presence of low concentrations of EGF, the receptor was exclusively internalized via CDE, whereas at high concentrations, receptors were equally distributed between CDE and CIE.
Saturation mechanisms (Zero-order ultrasensitivity) Zero-order ultrasensitivity takes place under saturating conditions. For example, consider an enzymatic step with a kinase, phosphatase, and substrate. Steady state levels of the phosphorylated substrate have an ultrasensitive response when there is enough substrate to saturate all available kinases and phosphatases. Under these conditions, small changes in the ratio of kinase to phosphatase activity can dramatically change the number of phosphorylated substrate (For a graph illustrating this behavior, see ). This enhancement in sensitivity of steady state phosphorylated substrate to Km, or the ratio of kinase to phosphatase activity, is termed zero-order to distinguish it from the first order behavior described by Michaelis-Menten dynamics, wherein the steady state concentration responds in a more gradual fashion than the switch-like behavior exhibited in ultrasensitivity. Using the notation from Goldbeter & Koshland, let W be a certain substrate protein and let W' be a covalently modified version of W. The conversion of W to W' is catalyzed by some enzyme E 1 {\displaystyle {\ce {E1}}} and the reverse conversion of W' to W is catalyzed by a second enzyme E 2 {\displaystyle {\ce {E2}}} according to following equations:
W + E 1 ⇌ d 1 a 1 W E 1 → k 1 W ′ + E 1
W ′ + E 2 ⇌ d 2 a 2 W ′ E 2 → k 2 W + E 2
{\displaystyle {\begin{array}{rcrcr}\\W+E_{1}&{\ce {<=>[a_1][d_1]}}&WE_{1}&{\ce {->[k_1]}}&W'+E_{1}\\{}\\W'+E_{2}&{\ce {<=>[a_2][d_2]}}&W'E_{2}&{\ce {->[k_2]}}&W+E_{2}\\{}\end{array}}}
The concentrations of all necessary components (such as ATP) are assumed to be constant and represented in the kinetic constants. Using the chemical equations above, the reaction rate equations for each component are:
d [ W ] d t = − a 1 [ W ] [ E 1 ] + d 1 [ W E 1 ] + k 2 [ W ′ E 2 ] {\displaystyle {\frac {d[W]}{dt}}=-a_{1}[W][E_{1}]+d_{1}[WE_{1}]+k_{2}[W'E_{2}]}
d [ W E 1 ] d t = a 1 [ W ] [ E 1 ] − ( d 1 + k 1 ) [ W E 1 ] {\displaystyle {\frac {d[WE_{1}]}{dt}}=a_{1}[W][E_{1}]-(d_{1}+k_{1})[WE_{1}]}
d [ W ′ ] d t = − a 2 [ W ′ ] [ E 2 ] + d 2 [ W ′ E 2 ] + k 1 [ W E 1 ] {\displaystyle {\frac {d[W']}{dt}}=-a_{2}[W'][E_{2}]+d_{2}[W'E_{2}]+k1[WE_{1}]}
d [ W ′ E 2 ] d t = a 2 [ W ′ ] [ E 2 ] − ( d 2 + k 2 ) [ W ′ E 2 ] {\displaystyle {\frac {d[W'E_{2}]}{dt}}=a_{2}[W'][E_{2}]-(d_{2}+k_{2})[W'E_{2}]}
The total concentration of each component is given by:
[ W T ] = [ W ] + [ W ′ ] + [ W E 1 ] + [ W ′ E 2 ] {\displaystyle [W_{T}]=[W]+[W']+[WE_{1}]+[W'E_{2}]}
[ E 1 T ] = [ E 1 ] + [ W E 1 ] {\displaystyle [E_{1T}]=[E_{1}]+[WE_{1}]}
[ E 2 T ] = [ E 2 ] + [ W ′ E 2 ] {\displaystyle [E_{2T}]=[E_{2}]+[W'E_{2}]}
The zero order mechanism assumes that the [ W T ] ≫ [ E 1 ] {\displaystyle [W_{T}]\gg [E_{1}]} or [ E 2 ] {\displaystyle [E_{2}]} . In other words, the system is in a Michaelis-Menten steady state, which means, to a good approximation, [ W E 1 ] {\displaystyle [WE_{1}]} and [ W ′ E 2 ] {\displaystyle [W'E_{2}]} are constant. From these kinetic expressions one can solve for V 1 / V 2 {\displaystyle V_{1}/V_{2}} at steady state defining W = [ W ] / [ W T ] {\displaystyle W=[W]/[W_{T}]} and W = 1 − W ′ {\displaystyle W=1-W'}
V 1 V 2 = W ′ ( 1 − W ′ + K 1 ) ( 1 − W ′ ) ( W ′ + K 2 ) , {\displaystyle {\frac {V_{1}}{V_{2}}}={\frac {W'\left(1-W'+K_{1}\right)}{\left(1-W'\right)\left(W'+K_{2}\right)}},}
where k 1 [ W E 1 ] = k 2 [ W ′ E 2 ] {\displaystyle k_{1}[WE_{1}]=k_{2}[W'E_{2}]} and
V 1 = k 1 [ E 1 T ] , {\displaystyle V_{1}=k_{1}[E_{1T}],}
V 2 = k 2 [ E 2 T ] , {\displaystyle V_{2}=k_{2}[E_{2T}],}
K 1 = d 1 + k 1 a 1 [ W T ] = K M 1 [ W T ] , {\displaystyle K_{1}={\frac {d_{1}+k_{1}}{a_{1}[W_{T}]}}={\frac {K_{M1}}{[W_{T}]}},}
K 2 = d 2 + k 2 a 2 [ W T ] = K M 2 [ W T ] . {\displaystyle K_{2}={\frac {d_{2}+k_{2}}{a_{2}[W_{T}]}}={\frac {K_{M2}}{[W_{T}]}}.}
When the V 1 / V 2 {\displaystyle V_{1}/V_{2}} is plotted against the molar ratio W ′ {\displaystyle W'} and W {\displaystyle W} it can be seen that the W to W' conversion occurs over a much smaller change in the V 1 / V 2 {\displaystyle V_{1}/V_{2}} ratio than it would under first order (non-saturating) conditions, which is the telltale sign of ultrasensitivity.
Positive Feedback Positive feedback loops can cause ultrasensitive responses. An example of this is seen in the transcription of certain eukaryotic genes in which non-cooperative transcription factor binding changes positive feedback loops of histone modification that results in an ultrasensitive activation of transcription. The binding of a transcription factor recruits histone acetyltransferases and methyltransferases. The acetylation and methylation of histones recruits more acetyltransferases and methyltransferases that results in a positive feedback loop. Ultimately, this results in activation of transcription. Additionally, positive feedback can induce bistability in Cyclin B1- by the two regulators Wee1 and Cdc25C, leading to the cell's decision to commit to mitosis. The system cannot be stable at intermediate levels of Cyclin B1, and the transition between the two stable states is abrupt when increasing levels of Cyclin B1 switches the system from low to high activity. Exhibiting hysteresis, for different levels of Cyclin B1, the switches from low to high and high to low states vary. However, the emergence of a bistable system is highly influenced by the sensitivity of its feedback loops. It has been shown in Xenopus egg extracts that Cdc25C hyperphosphorylation is a highly ultrasensitive function of Cdk activity, displaying a high value of the Hill coefficient (approx. 11), and the dephosphorylation step of Ser 287 in Cdc25C (also involved in Cdc25C activation) is even more ultrasensitive, displaying a Hill coefficient of approximately 32.
Allovalency A proposed mechanism of ultrasensitivity, called allovalency, suggests that activity "derives from a high local concentration of interaction sites moving independently of each other" Allovalency was first proposed when it was believed to occur in the pathway in which Sic1, is degraded in order for Cdk1-Clb (B-type cyclins) to allow entry into mitosis. Sic1 must be phosphorylated multiple times in order to be recognized and degraded by Cdc4 of the SCF Complex. Since Cdc4 only has one recognition site for these phosphorylated residues it was suggested that as the amount of phosphorylation increases, it exponentially increases the likelihood that Sic1 is recognized and degraded by Cdc4. This type of interaction was thought to be relatively immune to loss of any one site and easily tuned to any given threshold by adjusting the properties of individual sites. Assumptions for the allovalency mechanism were based on a general mathematical model that describes the interaction between a polyvalent disordered ligand and a single receptor site It was later found that the ultrasensitivity in Cdk1 levels by degradation of Sic1 is in fact due to a positive feedback loop.
Non-Zero-Order Ultrasensitivity in Membrane Proteins Modeling by Dushek et al. proposes a possible mechanism for ultrasensitivity outside of the zero-order regime. For the case of membrane-bound enzymes acting on membrane-bound substrates with multiple enzymatic sites (such as tyrosine-phosphorylated receptors like the T-Cell receptor), ultrasensitive responses could be seen, crucially dependent on three factors: 1) limited diffusion in the membrane, 2) multiple binding sites on the substrate, and 3) brief enzymatic inactivation following catalysis. Under these particular conditions, although the enzyme may be in excess of the substrate (first-order regime), the enzyme is effectively locally saturated with substrate due to the multiple binding sites, leading to switch-like responses. This mechanism of ultrasensitivity is independent of enzyme concentration, however the signal is significantly enhanced depending on the number of binding sites on the substrate. Both conditional factors (limited diffusion and inactivation) are physiologically plausible, but have yet to be experimentally confirmed. Dushek's modeling found increasing Hill cooperativity numbers with more substrate sites (phosphorylation sites), and with greater steric/diffusional hindrance between enzyme and substrate. This mechanism of ultrasensitivity based on local enzyme saturation arises partly from passive properties of slow membrane diffusion, and therefore may be generally applicable.
Dissipative Allostery The bacterial flagellar motor has been proposed to follow a dissipative allosteric model, where ultrasensitivity comes as a combination of protein binding affinity and energy contributions from the proton motive force (see Flagellar motors and chemotaxis below).
Impact of upstream and downstream components on module's ultrasensitivity In a living cell, ultrasensitive modules are embedded in a bigger network with upstream and downstream components. This components may constrain the range of inputs that the module will receive as well as the range of the module's outputs that network will be able to detect. Altszyler et al. (2014) studied how the effective ultrasensitivity of a modular system is affected by these restrictions. They found for some ultrasensitive motifs that dynamic range limitations imposed by downstream components can produce effective sensitivities much larger than that of the original module when considered in isolation.
Hill Coefficient Ultrasensitive behavior is typically represented by a sigmoidal curve, as small alterations in the stimulus [ L ] {\displaystyle [L]} can trigger large changes in the response θ {\displaystyle \theta } . One such relation is the Hill equation:
θ = [ L ] n K d + [ L ] n = [ L ] n ( K A ) n + [ L ] n , {\displaystyle \theta ={\frac {[L]^{n}}{K_{d}+[L]^{n}}}={\frac {[L]^{n}}{(K_{A})^{n}+[L]^{n}}},}
where n {\displaystyle n} is the Hill coefficient which quantifies the steepness of the sigmoidal stimulus-response curve and it is therefore a sensitivity parameter. It is often used to assess the cooperativity of a system. A Hill coefficient greater than one is indicative of positive cooperativity and thus, the system exhibits ultrasensitivity. Systems with a Hill coefficient of 1 are noncooperative and follow the classical Michaelis-Menten kinetics. Enzymes exhibiting noncooperative activity are represented by hyperbolic stimulus/response curves, compared to sigmoidal curves for cooperative (ultrasensitive) enzymes. In mitogen-activated protein kinase (MAPK) signaling (see example below), the ultrasensitivity of the signaling is supported by the sigmoidal stimulus/response curve that is comparable to an enzyme with a Hill coefficient of 4.0-5.0. This is even more ultrasensitive to the cooperative binding activity of hemoglobin, which has a Hill coefficient of 2.8.
Calculation From an operational point of view the Hill coefficient can be calculated as:
n H = log ( 81 ) log ( EC 90 / EC 10 ) {\displaystyle n_{H}={\frac {\ce {\log(81)}}{\ce {\log(EC90/EC10)}}}} . where EC 90 {\displaystyle {\ce {EC90}}} and EC 10 {\displaystyle {\ce {EC10}}} are the input values needed to produce the 10% and 90% of the maximal response, respectively.
Response Coefficient Global sensitivity measures such as the Hill coefficient do not characterise the local behaviours of the s-shaped curves. Instead, these features are well captured by the response coefficient measure defined as:
R ( x ) = x y d y d x {\displaystyle R(x)={\frac {x}{y}}{\frac {dy}{dx}}}
In systems biology, such system responses are referred to as control coefficients. Specifically, the concentration control coefficients measure the response of concentrations to changes in a given input. In addition, within the framework of the more general biochemical control