Hysteresis vs. Graded Responses: The Connections Make All the Difference

Science's STKE  11 May 2004:
Vol. 2004, Issue 232, pp. pe20
DOI: 10.1126/stke.2322004pe20


Biological regulatory systems have the potential to provide graded responses to stimuli or may demonstrate switch-like properties. Our understanding of the system design principles controlling these responses is still at a rudimentary stage, and here we consider several recent experimental and theoretical studies that focus on these system design principles. Overt positive feedback loops, or double-negative feedback loops, can produce bistable or multistable systems under the appropriate conditions and can produce graded responses under other conditions. Several design features favor bistability in negatively controlled genetic systems, including a high kinetic order for repression and a large difference in the rates of gene expression in the "on" and "off" states. In positive feedback, a high kinetic order for the activation of gene expression favors bistability. Multistability can result from the combined effects of positive and negative regulators, or from the combined effects of regulators that each demonstrate bistability. Finally, bistability can result in enzymatic systems in which multiple reversible covalent modifications occur, even when no overt feedback loops are present.

Even a cursory overview of biological systems reveals the presence of switches that show bistability or multistability. For example, such switches control cell fate decisions and development in metazoans. A multistable switch displays characteristic steady states with the absence of intermediate states; in the simplest case of bistability, the switch will display "on" and "off" states. A hallmark of such switches is hysteresis; that is, the response of the switch depends on its history. In the extreme case, a bistable switch shows irreversibility. Once a controlling stimulus reaches a certain threshold, the switch will change to an alternative steady state, which is then stably maintained even if the stimulus falls below the critical threshold. However, nature is also replete with examples of signaling systems that do not display hysteresis, but instead produce a graded output in response to variations in the controlling stimulus. Because both types of switches are constructed from similar components that utilize the same chemical mechanisms, it is apparent that the connectivity of the components is responsible for the behavior of the systems. That is, switch behavior is determined by the system design. With this realization, there has been considerable recent effort to elucidate the system design principles responsible for hysteresis and its prevention in biological systems.

The classic example of a bistability is the "preinduction effect" of the Escherichia coli lac operon, first noted by Monod and colleagues more than 50 years ago and studied in detail by Novick and Weiner in 1957 (1). These workers studied the induction of transcription of the lac operon by gratuitous inducers. (Gratuitous inducers are compounds that bind to and inactivate the repressor, yet are not metabolized by the induced enzymes.) The central observation was that cells that had been induced with a high concentration of a gratuitous inducer maintained full expression of the lac operon when shifted to medium containing a very low concentration of inducer, whereas cells that had never experienced the inducer showed no expression of the lac operon in the presence of the same very low concentration of inducer. Thus, lac operon expression was either "on" or "off," depending on the history of the cell.

The molecular basis of the cellular memory responsible for this behavior of the lac operon is the "autocatalytic" nature of the stimulus (Fig. 1). Use of gratuitous inducers bypasses the requirement for the internal synthesis of the natural allolactose inducer by the enzyme β-galactosidase, which is also responsible for destruction of allolactose. In the absence of control of internal inducer concentrations by β-galactosidase, the only relevant activity encoded by the lac operon is the LacY permease, which internalizes the gratuitous inducer. Because gratuitous induction increases the cellular concentration of permease molecules, which then enable further uptake of the inducer, an unrestrained positive feedback loop is present (Fig. 2F). The net effect of this positive feedback is that once the cell is penetrated by the first molecule of the gratuitous inducer, it very quickly reaches full activation, so that the cells essentially display all-or-none behavior. Upon cell division, each daughter acquires enough permease molecules to maintain full expression of the operon, even if the external concentration of the gratuitous inducer becomes very low.

Fig. 1.

Genetic system designs of the lac operon of E. coli during gratuitous induction (left) and induction by lactose (right). During gratuitous induction, the LacY permease facilitates the uptake of the inducer (depicted as IPTG; other gratuitous inducers have also been used that require LacY for efficient uptake). The inducer inactivates lac repressor (depicted as pink tetramer), which inhibits expression of the operon (depicted at the bottom). The lacZYA operon transcript is depicted as an arrow, and a heavy dashed line signifies the relationship between the lacY gene and its product, galactoside permease. The indirect inhibitory effects of glucose on inducer uptake and transcription initiation are depicted with dotted lines. In cells stimulated with lactose (right), the lacZ product β-galactosidase (depicted as blue tetramers) is required for the formation and breakdown of the internal inducer allolactose.

Fig. 2.

Gallery of selected bistable and oscillatory systems. (A) A simple bistable system of mutually positive influences. Each species increases the accumulation or activity of the other. (B) A simple bistable system of mutually negative influences. Each species inhibits the accumulation or activity of the other. This design results in toggle-switch behavior in E. coli (7). (C) Genetic toggle switch consisting of a species that activates its own expression with high cooperativity, and a repressing species (present at constant level) that blocks the positive feedback (8). The repressing species can exert control only when the activating species is at a low level. (D) Simple oscillator consisting of linked positive and negative influences (8). The system design is similar to that in (C), except that the activating species increases the accumulation or activity of the repressing species. This design is similar to a predator-prey relationship, where the repressing species corresponds to the predator. (E) Oscillatory system consisting of three repressors linked in a daisy chain (9). (F) Bistable system corresponding to gratuitous induction of the E. coli lac operon. Inducer (blue) inhibits the action of repressor (pink), which inhibits the accumulation of permease (green), which increases the accumulation of the inducer. (G) Simplified depiction of the positive feedback in the Mos-MEK-p42MAPK cascade. Mos (blue) increases the activity of MEK (pink), which increases the activity of p42 MAPK (green). The p42 MAPK increases the accumulation of Mos. (H) Simplified depiction of the mammalian circadian clock (10). The Clock::Bmal2 complex (blue) activates the expression of Period 2 (pink) and Cryptochrome (green). The latter inhibits activity of the Clock::Bmal2 complex and increases the accumulation of Per2, which increases activation of the Clock complex. (I) Bistability can arise from system-level effects in the absence of overt positive or negative feedback in systems in which multisite phosphorylation and dephosphorylation occur (5). An ordered system is depicted (although this is not necessary for bistability) where a kinase sequentially phosphorylates a species to produce singly and doubly phosphorylated species, and a phosphatase sequentially reverses the phosphorylations. As described in the text and (5), under certain conditions such systems display multiple steady states.

The lac operon does not display hysteresis or all-or-none behavior when induced with the physiological inducer lactose. Apparently, the destruction of the internal inducer allolactose by the induced product β-galactosidase serves as a sufficient brake on the positive feedback provided by LacY to convert the switch to one producing a graded output (Fig. 1). Indeed, this may explain the evolution of the curious mechanism by which both formation and breakdown of allolactose are catalyzed by one of the enzymes (β-galactosidase) encoded by the lac operon. Because the lac operon can be shifted between a bistable system and one producing a graded response simply by altering the nature of the inducer (gratuitous rather than metabolized), which essentially alters the system design (Fig. 1), it is readily apparent in this simple example that the system design is responsible for the switch behavior.

The description of the gratuitous induction of the lac operon provided above is a simplification that ignores several well-known aspects of lac regulation. These features include the role of external glucose in decreasing the concentration of adenosine 3′,5′-monophosphate (cAMP) within the cell, and the role of glucose in inhibiting the uptake of gratuitous inducer by inhibiting LacY (Fig. 1). Both of these effects of glucose are indirect and result from its control of the phosphorylation state of the phosphotransferase system component factor IIAglc; the unphosphorylated form of IIAglc inhibits the uptake of inducer by LacY, whereas phosphorylated IIAglc activates adenylyl cyclase. This combination of mechanisms is responsible for the catabolite repression of lac operon expression by glucose, although the precise contribution of each effect is difficult to dissect using classic experimental strategies. Ozbudak and colleagues (2) extended our understanding of this system by examining the relationship between glucose and a gratuitous inducer. They used a fluorescent reporter of lac operon expression to investigate the effects of glucose on the bistability of the system at the single-cell level. The presence of glucose did not eliminate the bistability of the system, but shifted to a higher range the external gratuitous inducer concentrations at which switching occurred. The intuitive explanation for this shift is that the presence of glucose does not alter the system design but only alters certain parameters, specifically, the rate of inducer uptake and the rate of transcription initiation of the unrepressed operon (Fig. 1). Thus, the potential for bistability is unaltered, but the condition at which it is observed is shifted.

The simple presence of a positive control loop does not ensure a bistable system. Additional conditions that must be present include a high kinetic order for the regulatory step (repression in the case of the lac system), and a sufficiently large difference in the rates of gene expression generated by the system in its "on" and "off" states (the repression ratio in the case of lac). Of these conditions, Ozbudak and colleagues (2) investigated the effect of the repression ratio, which was reduced by providing operator sites in multicopy, which served to titrate the repressor. Appealingly, when the repression ratio was reduced by a factor of ~35, the system was converted from a bistable system to one producing a graded response, in agreement with the model. Future experiments could further test the model by reducing the kinetic order of repression. The high kinetic order of lac repression is due to the cooperative binding of the tetrameric repressor to both the main lac operator (lacO1) and one of the two auxiliary lac operators found −84 bp upstream (lacO3) or +400 bp downstream (lacO2) from the transcription start site. This cooperative binding involves the formation of a DNA loop with the repressor simultaneously contacting lacO1 and either lacO2 or lacO3. Mutations that convert the tetrameric repressor to a dimer (by deleting its C-terminal tetramerization helix), or mutations that delete both auxillary operators, reduce the kinetic order of repression.

Setty and colleagues (3) also recently investigated the complex regulation of lac operon expression by using a fluorescent reporter system on a low-copy plasmid to assess the regulation of the system by exogenous isopropylthio-β-galactoside (IPTG) and cAMP. Their results indicate that promoter activity shows four different levels, depending on whether these effectors are both present at high concentrations, both at low concentrations, or one is high and the other is low. There were also four thresholds, two each for IPTG and cAMP, depending on whether the other effector was present or not. Thus, the different effectors (IPTG and cAMP) clearly display synergy, and the lac promoter logic gate does not correspond to a pure "and" gate (that is, there is not a requirement for both stimulatory factors). Modeling of the system suggested that the system could likely be converted to a purer "and" gate, or even to an "or" gate, with a small number of mutations. For example, strengthening LacI binding and weakening RNA polymerase binding was predicted to convert the system to a more pure "and" gate, whereas weakening LacI binding and strengthening cAMP-cyclic AMP receptor protein (CRP) binding is predicted to convert the system to an "or" gate. Future work should experimentally address these predictions. Because a small number of mutations are predicted to convert the lac promoter to a more pure "and" gate, we must assume that the complex natural system provides a selective advantage that would not result from a pure "and" gate under some environmental conditions.

How can one know whether a positive feedback system has the potential for bistability? Angeli and colleagues (4) argue that the properties of the system under conditions in which the feedback is eliminated may suggest the potential for bistability. Specifically, any monotone system (that is, a positive control system without any negative regulatory loops) with sufficient cooperativity (or gain) in the absence of positive feedback should show bistability in the presence of positive feedback. Put simply, the inherent gain of the system in combination with the positive feedback provides the basis for bistability. Because it is frequently possible to eliminate positive feedback loops experimentally, the method describes a feasible approach to ascertain the potential for bistability in complex regulatory systems, even if all components and regulatory interactions have not been defined.

A number of studies have demonstrated that a positive feedback loop or an even number of linked negative feedback loops may give rise to bistable genetic systems, as intuition would suggest. Markevich and colleagues (5) demonstrated that bistability can result in systems with multisite phosphorylation and dephosphorylation, such as cellular mitogen-activated protein kinase (MAPK) cascades, without overt positive or double-negative feedback loops--a nonintuitive result that may have considerable physiological significance. The activity of MAPK is controlled by phosphorylation at two sites (by a MAPK kinase) and dephosphorylation [by MAPK phosphatase (MKP)]. Ordered systems for the two kinase and two phosphatase steps [in which one of the two sites is always phosphorylated first and the other site is always dephosphorylated first (Fig. 2I)], and partially ordered systems and random-order systems, were considered with similar results. Bistability in such systems results from the indirect, "system-level" effects of the species on the rates of interconversion, and depends on certain conditions: both phosphorylation and dephosphorylation reactions must be distributive (nonprocessive, that is, the product is released from the enzyme before it is bound again and modified a second time); at least one of the enzymes is saturated with its substrate (the Km is well below the substrate concentration); and the substrate of the first step can competitively inhibit the second modification step. Markevich and colleagues demonstrate that, with the appropriate ratios of Km for the various species and the appropriate ratios of catalytic activities of the enzymes, such systems display bistability. For example, consider the situation depicted in Fig. 2I in which a kinase and phosphatase bring about the double phosphorylation and dephosphorylation of a target protein; consider also the case in which the levels of the three proteins are fixed. When the system starts from a state in which the substrate is completely dephosphorylated, the production of the doubly phosphorylated species is very difficult, because the kinase is saturated with the unphosphorylated form of the substrate. Because the doubly phosphorylated species is not present, it cannot prevent dephosphorylation of singly phosphorylated species. Similarly, when the system starts from a state in which the substrate is doubly phosphorylated, the production of the completely dephosphorylated species is very difficult, because the phosphatase is saturated with the doubly phosphorylated form of the protein, and the unphosphorylated form is not present and thus cannot inhibit the phosphorylation of the singly phosphorylated species.

Estimates based on reported values of the Km and kcat for the phosphorylation and dephosphorylation reactions of an MAPK cascade suggest that bistability is feasible in vivo. Interestingly, bistability is also predicted in systems in which multiple phosphorylation events are catalyzed by distinct enzymes, as long as the dephosphorylation steps are catalyzed by a single phosphatase. Many cellular signaling systems involving reversible and multiple phosphorylations may therefore display "intrinsic" bistability in the absence of an overt feedback loop.

Markevich and colleagues also note that the "intrinsic" bistability of enzymatic systems with the properties discussed above can act in concert with overt positive feedback loops to create multistability (5). For example, cascades involving multiple phosphorylation steps (such as MAPK cascades) are regulated by overt positive feedback loops. For such systems, both mechanisms of hysteresis may be operative if the kinetic parameters are appropriate, resulting in multistability.

The above discussion notwithstanding, recent results from Xiong and Ferrell (6) demonstrate the importance of positive feedback loops in the maturation of Xenopus oocytes. This maturation is promoted by progesterone, which activates an MAPK cascade consisting of Mos, MAPK kinase (MEK), and p42 MAPK. This cascade leads to the activation of cyclin B-Cdc2, in part by inhibition of the Cdc2 inhibitor Myt1. The active Cdc2 also promotes its own activation by activating its activator, Cdc25, and has a positive influence on MAPK by promoting the accumulation of its activator, Mos. Finally, the MAPK branch of the regulatory system displays an independent positive control loop, as the active p42 MAPK also promotes the accumulation of Mos. Thus, the system contains at least two direct positive control loops and two indirect positive control loops. At least two of these loops (those affecting Mos accumulation) require protein synthesis; this feature formed the basis of an experimental strategy to test the importance of the positive feedback loops.

Oocyte maturation displays a characteristic "maturation inertia," in which only transient treatment with the inducing hormone is required to produce a sustained response. After transient progesterone treatment of Xenopus oocytes, p42 MAPK and Cdc2 activities remain high even after the cells are washed free of progesterone. Thus, these activities display hysteresis. The hysteresis seems to be a feature of the Cdc2-p42MAPK system itself and is not due to factors upstream of Mos, as demonstrated by the introduction of an activating stimulus from a fusion protein consisting of the protein kinase Raf and the estrogen receptor; this fusion protein permits stimulation of the system under the control of estrogen, in the absence of progesterone. Transient stimulation by estrogen resulted in hysteretic activation of p42 MAPK and Cdc2, as was observed with progesterone. Thus, the mechanism of stimulation seems irrelevant, and the hysteresis of the system seems to be due to the MAPK-Cdc2 system design.

The importance of the positive feedback loops controlling Mos accumulation was tested in several ways, most notably by inhibiting protein synthesis with cyclohexamide and by specifically preventing Mos expression with an antisense oligonucleotide. Both treatments eliminated the irreversibility of p42 MAPK and Cdc2 activation by hormonal stimulation. Thus, the positive feedback loops controlling Mos accumulation play a central role in the irreversibility of the response to hormonal stimulation.

Understanding the behavior of complex biological systems requires a two-pronged approach. First, we need to know the circuitry, that is, the system designs that are present in nature. This problem can be approached only experimentally, by identifying the relevant components, the chemistry of their interactions, the mechanisms of the regulatory interactions, and the relevant kinetic parameters. Second, we need to decipher the rules governing system-level behaviors, that is, the design principles responsible for various system behaviors. The recent articles highlighted in this perspective show that exciting progress is occurring on both fronts.


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