Multistep Mechanisms

Multipstep ultrasensitivity occurs when a single effector acts on several steps in a cascade.^[18] 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.^[19] Cascade ultrasensitivity can reduce noise and therefore require less input for activation.^[12] 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.^[20]

Buffering Mechanisms

Buffering Mechanisms such as molecular titration can generate ultrasensitivity. In vitro, this can be observed for the simple mechanism:

${\displaystyle {\ce {A + B <=> AB}}}$

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 ${\displaystyle B_{T}}$ ( = [B] + [AB]) is much greater than the ${\displaystyle K_{d}}$, this system exhibits a threshold determined by the concentration of ${\displaystyle B_{T}}$.^[21] At concentrations of ${\displaystyle A_{T}}$ ( = [A] +[AB]), lower than ${\displaystyle B_{T}}$, B acts as a buffer to free A and nearly all A will be found as AB. However, at the equivalence point, when ${\displaystyle A_{T}}$ ≈ ${\displaystyle B_{T}}$, ${\displaystyle B_{T}}$ can no longer buffer the increase in ${\displaystyle A_{T}}$, so a small increase in ${\displaystyle A_{T}}$ causes a large increase in A.^[22] The strength of the ultrasensitivity of [A] to changes in ${\displaystyle A_{T}}$ is determined by ${\displaystyle B_{T}}$/${\displaystyle K_{d}}$.^[22] Ultrasensitivity occurs when this ratio is greater than one and is increased as the ratio increases. Above the equivalence point, ${\displaystyle A_{T}}$ 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.^[22]

One example of a buffering mechanism is protein sequestration, which is a common mechanism found in signalling and regulatory networks.^[23] 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

Saturation mechanisms (Zero-order ultrasensitivity)

Zero-order ultrasensitivity takes place under saturating conditions.^[25] 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.^[25][26] 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 ^[5]). 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.^[18]

Using the notation from Goldbeter & Koshland,^[5] 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 ${\displaystyle {\ce {E1}}}$ and the reverse conversion of W' to W is catalyzed by a second enzyme ${\displaystyle {\ce {E2}}}$ according to following equations:

${\displaystyle {\begin{array}{rcrcr}\\W+E_{1}&{\ce {<=>[a_1][d_1]}}&WE_{1}&{\ce {->[k_1]}}&W'+E_{1}\\{}\\W'+E_{2}&{\ce {<=>[a_2][d_2]}}&W'E_{2}&{\ce {->[k_2]}}&W+E_{2}\\{}\end{array}}}$

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:

${\displaystyle {\frac {d[W]}{dt}}=-a_{1}[W][E_{1}]+d_{1}[WE_{1}]+k_{2}[W'E_{2}]}$

${\displaystyle {\frac {d[WE_{1}]}{dt}}=a_{1}[W][E_{1}]-(d_{1}+k_{1})[WE_{1}]}$

${\displaystyle {\frac {d[W']}{dt}}=-a_{2}[W'][E_{2}]+d_{2}[W'E_{2}]+k1[WE_{1}]}$

${\displaystyle {\frac {d[W'E_{2}]}{dt}}=a_{2}[W'][E_{2}]-(d_{2}+k_{2})[W'E_{2}]}$

The total concentration of each component is given by:

${\displaystyle [W_{T}]=[W]+[W']+[WE_{1}]+[W'E_{2}]}$

${\displaystyle [E_{1T}]=[E_{1}]+[WE_{1}]}$

${\displaystyle [E_{2T}]=[E_{2}]+[W'E_{2}]}$

The zero order mechanism assumes that the ${\displaystyle [W_{T}]\gg [E_{1}]}$ or ${\displaystyle [E_{2}]}$. In other words, the system is in a Michaelis-Menten steady state, which means, to a good approximation, ${\displaystyle [WE_{1}]}$ and ${\displaystyle [W'E_{2}]}$ are constant. From these kinetic expressions one can solve for ${\displaystyle V_{1}/V_{2}}$ at steady state defining ${\displaystyle W=[W]/[W_{T}]}$ and ${\displaystyle W=1-W'}$

${\displaystyle {\frac {V_{1}}{V_{2}}}={\frac {W'\left(1-W'+K_{1}\right)}{\left(1-W'\right)\left(W'+K_{2}\right)}},}$

where ${\displaystyle k_{1}[WE_{1}]=k_{2}[W'E_{2}]}$ and

${\displaystyle V_{1}=k_{1}[E_{1T}],}$

${\displaystyle V_{2}=k_{2}[E_{2T}],}$

${\displaystyle K_{1}={\frac {d_{1}+k_{1}}{a_{1}[W_{T}]}}={\frac {K_{M1}}{[W_{T}]}},}$

${\displaystyle K_{2}={\frac {d_{2}+k_{2}}{a_{2}[W_{T}]}}={\frac {K_{M2}}{[W_{T}]}}.}$

When the ${\displaystyle V_{1}/V_{2}}$ is plotted against the molar ratio ${\displaystyle W'}$ and ${\displaystyle W}$ it can be seen that the W to W' conversion occurs over a much smaller change in the ${\displaystyle V_{1}/V_{2}}$ 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.^[17]

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.^[27] 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.^[28]

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"^[29] 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.^[30] 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^[29] It was later found that the ultrasensitivity in Cdk1 levels by degradation of Sic1 is in fact due to a positive feedback loop.^[31]

Non-Zero-Order Ultrasensitivity in Membrane Proteins

Modeling by Dushek et al.^[32] 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.^[32] 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).

Calculation

From an operational point of view the Hill coefficient can be calculated as:

${\displaystyle n_{H}={\frac {\ce {\log(81)}}{\ce {\log(EC90/EC10)}}}}$.

where ${\displaystyle {\ce {EC90}}}$ and ${\displaystyle {\ce {EC10}}}$ are the input values needed to produce the 10% and 90% of the maximal response, respectively.

Supramultiplicativity

Several authors have reported the existence of supramultiplicative behavior in signaling cascades ^[40][33](i.e. the ultrasensitivity of the combination of layers is higher than the product of individual ultrasensitivities), but in many cases the ultimate origin of supramultiplicativity remained elusive. Altszyler et al. (2017)^[37] framework naturally suggested a general scenario where supramultiplicative behavior could take place. This could occur when, for a given module, the corresponding Hill's input working range was located in an input region with local ultrasensitivities higher than the global ultrasensitivity of the respective dose-response curve.

MAP Kinase Signaling Cascade

A ubiquitous signaling motif that exhibits ultrasensitivity is the MAPK (mitogen-activated protein kinase) cascade, which can take a graded input signal and produce a switch-like output, such as gene transcription or cell cycle progression. In this common motif, MAPK is activated by an earlier kinase in the cascade, called MAPK kinase, or MAPKK. Similarly, MAPKK is activated by MAPKK kinase, or MAPKKK. These kinases are sequentially phosphorylated when MAPKKK is activated, usually via a signal received by a membrane-bound receptor protein. MAPKKK activates MAPKK, and MAPKK activates MAPK.^[35] Ultrasensitivity arises in this system due to several features:

1. MAPK and MAPKK both require two separate phosphorylation events to be activated.
2. The reversal of MAPK phosphorylation by specific phosphatases requires an increasing concentration of activation signals from each prior kinase to achieve an output of the same magnitude.
3. The MAPKK is at a concentration above the K_Μ for its specific phosphatase and MAPK is at a concentration above the K_Μ for MAPKK.

Ultrasensitivity in the mitogen-activated protein (MAP) kinase cascade^[35]

Besides the MAPK cascade, ultrasensitivity has also been reported in muscle glycolysis, in the phosphorylation of isocitrate dehydrogenase and in the activation of the calmodulin-dependent protein kinase II (CAMKII).^[34]

An ultrasensitive switch has been engineered by combining a simple linear signaling protein (N-WASP) with one to five SH3 interaction modules that have autoinhibitory and cooperative properties. Addition of a single SH3 module created a switch that was activated in a linear fashion by exogenous SH3-binding peptide. Increasing number of domains increased ultrasensitivity. A construct with three SH3 modules was activated with an apparent Hill coefficient of 2.7 and a construct with five SH3 module was activated with an apparent Hill coefficient of 3.9.^[41]

Translocation

During G2 phase of the cell cycle, Cdk1 and cyclin B1 makes a complex and forms maturation promoting factor (MPF). The complex accumulates in the nucleus due to phosphorylation of the cyclin B1 at multiple sites, which inhibits nuclear export of the complex. Phosphorylation of Thr19 and Tyr15 residues of Cdk1 by Wee1 and MYT1 keeps the complex inactive and inhibits entry into mitosis whereas dephosphorylation of Cdk1 by CDC25C phosphatase at Thr19 and Tyr15 residues, activates the complex which is necessary in order to enter mitosis. Cdc25C phosphatase is present in the cytoplasm and in late G2 phase it is translocated into the nucleus by signaling such as PIK1,^[42] PIK3.^[43] The regulated translocation and accumulation of the multiple required signaling cascade components, MPF and its activator Cdc25, in the nucleus generates efficient activation of the MPF and produces switch-like, ultrasensitive entry into mitosis.^[4]

The figure^[4] shows different possible mechanisms for how increased regulation of the localization of signaling components by the stimulus (input signal) shifts the output from Michaelian response to ultrasensitive response. When stimulus is regulating only inhibition of Cdk1-cyclinB1 nuclear export, the outcome is Michaelian response, Fig (a). But if the stimulus can regulate localization of multiple components of the signaling cascade, i.e. inhibition of Cdk1-cyclinB1 nuclear export and translocation of the Cdc25C to nucleus, then the outcome is ultrasensitive response, Fig (b). As more components of the signaling cascade are regulated and localized by the stimulus—i.e. inhibition of Cdk1-cyclinB1 nuclear export, translocation of the Cdc25C to the nucleus, and activation of Cdc25C—the output response becomes more and more ultrasensitive, Fig(c).^[4]

Buffering (decoy)

During mitosis, mitotic spindle orientation is essential for determining the site of cleavage furrowing and position of daughter cells for subsequent cell fate determination.^[44] This orientation is achieved by polarizing cortical factors and rapid alignment of the spindle with the polarity axis. In fruit flies, three cortical factors have been found to regulate the position of the spindle: heterotrimeric G protein α subunit (Gαi),^[45] Partner of Inscuteable (Pins),^[46] and Mushroom body defect (Mud).^[47] Gαi localizes at apical cortex to recruit Pins. Upon binding to GDP-bound Gαi, Pins is activated and recruits Mud to achieve polarized distribution of cortical factors.^[48] N-terminal tetratricopeptide repeats (TPRs) in Pins is the binding region for Mud, but is autoinhibited by intrinsic C-terminal GoLoco domains (GLs) in the absence of Gαi.^[49][50] Activation of Pins by Gαi binding to GLs is highly ultrasensitive and is achieved through the following decoy mechanism:^[14] GLs 1 and 2 act as a decoy domains, competing with the regulatory domain, GL3, for Gαi inputs. This intramolecular decoy mechanism allows Pins to establish its threshold and steepness in response to distinct Gαi concentration. At low Gαi inputs, the decoy GLs 1 and 2 are preferentially bound. At intermediate Gαi concentration, the decoys are nearly saturated, and GL3 begins to be populated. At higher Gαi concentration, the decoys are fully saturated and Gαi binds to GL3, leading to Pins activation. Ultrasensitivity of Pins in response to Gαi ensures that Pins is activated only at the apical cortex where Gαi concentration is above the threshold, allowing for maximal Mud recruitment.

Switching Behavior of GTPases

GTPases are enzymes capable of binding and hydrolyzing guanosine triphosphate (GTP). Small GTPases, such as Ran and Ras, can exist in either a GTP-bound form (active) or a GDP-bound form (inactive), and the conversion between these two forms grants them a switch-like behavior.^[51] As such, small GTPases are involved in multiple cellular events, including nuclear translocation and signaling.^[52] The transition between the active and inactive states is facilitated by guanine nucleotide exchange factors (GEFs) and GTPase activating proteins (GAPs).^[53]

Computational studies on the switching behavior of GTPases have revealed that the GTPase-GAP-GEF system displays ultrasensitivity.^[54] In their study, Lipshtat et al. simulated the effects of the levels of GEF and GAP activation on the Rap activation signaling network in response to signals from activated α2-adrenergic (α2R) receptors, which lead to degradation of the activated Rap GAP. They found that the switching behavior of Rap activation was ultrasensitive to changes in the concentration (i.e. amplitude) and the duration of the α2R signal, yielding Hill coefficients of nH=2.9 and nH=1.7, respectively (a Hill coefficient greater than nH=1 is characteristic of ultrasensitivity ^[55]). The authors confirmed this experimentally by treating neuroblasts with HU-210, which activates RAP through degradation of Rap GAP. Ultrasensitivity was observed both in a dose-dependent manner (nH=5±0.2), by treating cells with different HU-210 concentrations for a fixed time, and in a duration-dependent manner (nH=8.6±0.8), by treating cells with a fixed HU-210 concentration during varying times.

By further studying system, the authors determined that (the degree of responsiveness and ultrasensitivity) was heavily dependent on two parameters: the initial ratio of kGAP/kGEF, where the k's incorporate both the concentration of active GAP or GEF and their corresponding kinetic rates; and the signal impact, which is the product of the degradation rate of activated GAP and either the signal amplitude or the signal duration.^[54] The parameter kGAP/kGEF affects the steepness of the transition from the two states of the GTPase switch, with higher values (~10) leading to ultrasensitivity. The signal impact affects the switching point. Therefore, by depending on the ratio of concentrations rather than on individual concentrations, the switch-like behavior of the system can also be displayed outside of the zero-order regime.

Ultrasensitivity and Neuronal Potentiation

Persistent stimulation at the neuronal synapse can lead to markedly different outcomes for the post-synaptic neuron. Extended weak signaling can result in long-term depression (LTD), in which activation of the post-synaptic neuron requires a stronger signal than before LTD was initiated. In contrast, long-term potentiation (LTP) occurs when the post-synaptic neuron is subjected to a strong stimulus, and this results in strengthening of the neural synapse (i.e., less neurotransmitter signal is required for activation).

In the CA1 region of the hippocampus, the decision between LTD and LTP is mediated solely by the level of intracellular ${\displaystyle {\ce {\scriptstyle Ca^{2}+}}}$ at the post-synaptic dendritic spine. Low levels of ${\displaystyle {\ce {\scriptstyle Ca^{2}+}}}$ (resulting from low-level stimulation) activates the protein phosphatase calcineurin, which induces LTD. Higher levels of ${\displaystyle {\ce {\scriptstyle Ca^{2}+}}}$ results in activation of ${\displaystyle \scriptstyle \color {Blue}{\ce {Ca^2+}}}$/calmodulin-dependent protein kinase II (CaMKII), which leads to LTP. The difference in Ca²⁺ concentration required for a cell to undergo LTP is only marginally higher than for LTD, and because neurons show bistability (either LTP or LTD) following persistent stimulation, this suggests that one or more components of the system respond in a switch-like, or ultrasensitive manner. Bradshaw et al. demonstrated that CaMKII (the LTP inducer) responds to intracellular calcium levels in an ultrasensitive manner, with <10% activity at 1.0 uM and ~90% activity at 1.5 uM, resulting in a Hill coefficient of ~8. Further experiments showed that this ultrasenstivity was mediated by cooperative binding of CaMKII by two molecules of calmodulin (CaM), and autophosphorylation of activated CaMKII leading to a positive feedback loop.^[56]

In this way, intracellular calcium can induce a graded, non-ultrasensitive activation of calcineurin at low levels, leading to LTD, whereas the ultrasensitive activation of CaMKII results in a threshold intracellular calcium level that generates a positive feedback loop that amplifies the signal and leads to the opposite cellular outcome: LTP. Thus, binding of a single substrate to multiple enzymes with different sensitivities facilitates a bistable decision for the cell to undergo LTD or LTP.

Ultrasensitivity in Development

It has been suggested that zero-order ultrasensitivity may generate thresholds during development allowing for the conversion of a graded morphogen input to a binary switch-like response.^[57] Melen et al. (2005) have found evidence for such a system in the patterning of the Drosophila embryonic ventral ectoderm.^[58] In this system, graded mitogen activated protein kinase (MAPK) activity is converted to a binary output, the all-or-none degradation of the Yan transcriptional repressor. They found that MAPK phosphorylation of Yan is both essential and sufficient for Yan's degradation. Consistent with zero-order ultrasensitivity an increase in Yan protein lengthened the time required for degradation but had no effect on the border of Yan degradation in developing embryos. Their results are consistent with a situation where a large pool of Yan becomes either completely degraded or maintained. The particular response of each cell depends on whether or not the rate of reversible Yan phosphorylation by MAPK is greater or less than dephosphorylation. Thus, a small increase in MAPK phosphorylation can cause it to be the dominant process in the cell and lead to complete degradation of Yan.

Multistep-feedback loop mechanism also leads to ultrasensitivity

Multistep-feedback loop mechanism also leads to ultrasensitivity. There is paper introducing that engineering synthetic feedback loops using yeast mating mitogen-activated protein (MAP) kinase pathway as a model system.

In Yeast mating pathway: alpha-factor activates receptor, Ste2, and Ste4 and activated Ste4 recruits Ste5 complex to membrane, allowing PAK-like kinase Ste20 (membrane-localized) to activate MAPKKK Ste11. Ste11 and downstream kinases, Ste7 (MAPKK) and Fus3 (MAPK), are colocalized on the scaffold and activation of cascade leads to transcriptional program. They used pathway modulators outside of core cascade, Ste50 promotes activation of Ste11 by Ste20; Msg5 (negative, red) is MAPK phosphatase that deactivates Fus3 (Fig.2A).

What they built was circuit with enhanced ultrasensitive switch behavior by constitutively expressing a negative modulator, Msg5 which is one of MAPK phosphatase and inducibly expressing a positive modulator, Ste50 which is pathway modulators outside of core cascade(Fig.2B). The success of this recruitment-based engineering strategy suggests that it may be possible to reprogram cellular responses with high precision.^[59]

Flagellar motors and chemotaxis

The rotational direction of E. coli is controlled by the flagellar motor switch. A ring of 34 FliM proteins around the rotor bind CheY, whose phosphorylation state determines whether the motor rotates in a clockwise or counterclockwise manner. The rapid switching mechanism is attributed to an ultrasensitive response, which has a Hill coefficient of ~10. This system has been proposed to follow a dissipative allosteric model, in which rotational switching is a result of both CheY binding and energy consumption from the proton motive force, which also powers the flagellar rotation.^[60]

Limitations in Modularity

It has been shown that the integration of a given synthetic ultrasensitive module with upstream and downstream components often alters its information-processing capabilities.^[33] This effects must be taken into account in the design process.