Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations

Transcriptional repression can occur via various mechanisms, such as blocking, sequestration and displacement. For instance, the repressors can hold the activators to prevent binding with DNA or can bind to the DNA-bound activators to block their transcriptional activity. Although the transcription can be completely suppressed with a single mechanism, multiple repression mechanisms are used together to inhibit transcriptional activators in many systems, such as circadian clocks and NF-κB oscillators. This raises the question of what advantages arise if seemingly redundant repression mechanisms are combined. Here, by deriving equations describing the multiple repression mechanisms, we find that their combination can synergistically generate a sharply ultrasensitive transcription response and thus strong oscillations. This rationalizes why the multiple repression mechanisms are used together in various biological oscillators. The critical role of such combined transcriptional repression for strong oscillations is further supported by our analysis of formerly identified mutations disrupting the transcriptional repression of the mammalian circadian clock. The hitherto unrecognized source of the ultrasensitivity, the combined transcriptional repressions, can lead to robust synthetic oscillators with a previously unachievable simple design.


Introduction
Transcription, the first step of gene expression, is regulated by activators and repressors, i.e. the bindings of the activators and repressors to a specific DNA sequence promote and downregulate transcription, respectively [1,2]. The repressors can also indirectly inhibit transcription by binding with the activators rather than with DNA (figure 1a) [3,4]. That is, the repressors can bind to the DNA-bound activators to block their transcriptional activity (blocking; figure 1a), hold the activators to prevent them from binding with DNA (sequestration; figure 1a), and dissociate the activators from DNA by forming a complex (displacement; figure 1a).
Each repression mechanism appears to be able to suppress transcription solely. However, various repressors use a combination of multiple repression mechanisms [3]. For example, retinoblastoma (Rb) protein, a key regulator of mammalian cell cycle genes, represses transcription by blocking the activator and recruiting histone deacetylase, which alters the structure of chromatin [5][6][7]. Similarly, PHO80, a component of a yeast nutrient-responsive signalling pathway, represses transcription by blocking the activator and sequestering the activator in the cytoplasm [8][9][10]. This raises the question of the advantages of using a combination of multiple repression mechanisms, which seems redundant.
In the transcriptional negative feedback loop (NFL) of various biological oscillators, repressors also inhibit their own transcriptions via combinations of the multiple repression mechanisms. For example, IκBα inhibits its own transcriptional activator NF-κB by sequestering it in the cytoplasm [11] as well as displacing it from DNA [12,13], which induces the NF-κB oscillation under stress conditions. In the transcriptional NFL of the circadian clock, the transcription is also suppressed in multiple ways. Specifically, in the Drosophila circadian clock, the repressor (PER : TIM) sequesters its own transcriptional activator (CLK:CYC) from DNA (sequestration), blocks the transcriptional activity by binding to DNA-bound CLK:CYC (blocking), and then displaces it from DNA (displacement; figure 1b) [14]. Similarly, in the mammalian circadian clock, the repressors (PER:CRY and CRY) also inhibit their own transcriptional activator (CLOCK:BMAL1) by sequestration, blocking and displacement (figure 1c) [15][16][17].
The transcriptional NFL can generate oscillations when the transcriptional activity shows an ultrasensitive response to changes in the concentration of repressors [18][19][20][21]. Such ultrasensitivity can be generated solely by sequestration when the activators and repressors tightly bind [22,23]. In particular, the sequestration requires only tight binding, which seems to be physiologically more achievable than the conditions for the other ultrasensitivity-generating mechanisms based on cooperativity (e.g. cooperative oligomerization). Thus, sequestration has recently been adopted for mathematical models of circadian clocks [21,[24][25][26][27][28][29][30][31]. However, Heidebrecht et al. pointed out that the tightness of the binding between the activator and repressor required for the sequestration to generate sustained rhythms is beyond the physiologically plausible binding affinity [32].
Here, we find that combining multiple transcriptional repression mechanisms can synergistically generate ultrasensitivity by deriving their governing equations. Specifically, we find that the sole blocking-type repression can generate only low-sensitivity transcriptional activity. When sequestration is added, the ultrasensitivity can be generated with stronger sequestration compared to the blocking. The required strong sequestration is challenging to achieve with a physiologically plausible binding affinity. Interestingly, this limitation to generate ultrasensitivity and strong oscillations can be overcome by adding displacement. To test whether the combination of the multiple repressions is critical for the mammalian circadian clock to generate strong rhythms, we investigated the previously identified mutations disrupting the transcriptional repressions [33][34][35][36]. Indeed, when any of the blocking, sequestration or displacement was disrupted, the circadian rhythms of PER2-LUC became weaker in mice. Our work explains why the combination of seemingly redundant repression mechanisms is used in various systems requiring ultrasensitivity, such as the cell cycle and the circadian clock.

The sole blocking-type repression generates a hyperbolic response in the transcriptional activity
To investigate how the transcription is regulated by the multiple repression mechanisms (figure 1a), we first constructed a model describing the single blocking-type repression (figure 2a; see Methods for details). In the model, the transcription is triggered when the free activator (A) binds to the free DNA (E F ) with a dissociation constant of K a to form the activated DNA (E A ). The transcription is inhibited when the repressor (R) binds to the DNA-bound A (E A ) to form ternary complex (E R ) with a dissociation constant of K b (i.e. the blocking-type repression). Therefore, the transcriptional activity is proportional to the probability that DNA is bound with only A and not R, i.e.  10 -4 10 -1 10 -7 10 -1 effective Hill coefficient 10 2  Figure 2. The combination of multiple repression mechanisms leads to ultrasensitive transcription response. (a) Diagram of the model describing the blocking-type repression. The binding of the activator (A) to DNA with a dissociation constant of K a leads to the transcription, and the binding of the repressor (R) to the DNAbound A with a dissociation constant of K b , inhibits the transcription. (b) As the molar ratio between the total repressor and activator concentrations (R T ) increases or their binding affinity increases (i.e.K b decreases), the transcriptional activity decreases. The sensitivity of the transcriptional activity is quantified using the effective Hill coefficient (Log(81)=Log(EC10=EC90)), which increases as the width of the EC90 and EC10 box decreases (i.e. the red box). The grey dashed lines denote the 10% and 90% values of the maximal transcriptional activity, respectively. Here,K a ¼ 10 À4 . (c) The effective Hill coefficient is one regardless of the values ofK b andK a , indicating that the sole blocking can generate only low sensitivity. The square and triangle marks represent the parameter values used for (b). (d ) The sequestration-type repression is added to the blocking model in (a): R sequesters the free A with a dissociation constant of K s from DNA. (e) When the sequestration is weaker than the blocking (i.e.K s .K b ; dotted line), the sensitivity of the transcriptional activity is similar to that regulated by only the blocking-type repression (b). On the other hand, when the sequestration is stronger than the blocking (i.e.K s ,K b ; solid and dashed lines), a switch-like transition in the transcriptional activity occurs. Here,K b ¼ 10 À5 andK a ¼ 10 À4 . (f ) The effective Hill coefficient increases asK s decreases. The circle, square and triangle marks represent the parameter values used for (e). (g) The displacement-type repression is added to the model in (d): the R A complex dissociates from DNA with a dissociation constant of K d . (h) WhenK d ¼K a ¼ 10 À4 (dashed line), it satisfies the detailed balance condition (i.e.K sK d =K bK a ¼ 1) and thus the displacement has no effect on the transcriptional activity (cf. dashed line in (e)). When the effective displacement occurs (i.e.K d .K a ; solid line), the sensitivity increases. Here,K s ¼ 10 À5 ,K b ¼ 10 À5 andK a ¼ 10 À4 . (i) WhenK d .K a , the effective Hill coefficients become larger compared to those obtained with the sequestration and blocking (f ). The circle, square and triangle marks represent the parameter values used for (h).
royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20210084 to one, the transcriptional activity and E A =E T become the same. Thus, for simplicity, we refer to E A =E T as the transcriptional activity throughout this study.
The transcriptional activity (E A =E T ) increases as A increases or R decreases. This relationship can be quantified by deriving the steady state of E A =E T . Because the steady state of E F depends on the single pair of binding and unbinding reactions with the dissociation constant of K a , its steady state equation is AE F ¼ K a E A . Similarly, the steady state equation of E R is also simple as RE A ¼ K b E R . Therefore, E F : E A : E R = 1: A/K a : ðR=K b ÞðA=K a Þ at the steady state, leading to the steady state of E A =E T as follows: where A and R are the steady states of the free activator and repressor, respectively (see Methods for details).
Because the steady states of A and R depend on the dissociation constants (i.e. K a and K b ), it is challenging to analyse equation (2.1). Equation (2.1) can be further simplified because the concentration of DNA is typically negligible compared to the concentration of activator and repressor proteins (see Methods for details about the validity of the assumption). Specifically, E A and E R can be neglected in the conserved total concentration of the activator ðA T ¼ A þ E A þ E R Þ and the repressor ðR T ¼ R þ E R Þ, and thus A % A T and R % R T . This allows us to get the simplified approximation for equation (2.1) as follows: , ð2:2Þ whereR T ¼ R T =A T is the molar ratio between R T and A T , are the dissociation constants normalized by the concentration of the total activator. Equation (2.2) indicates that the transcriptional activity shows a hyperbolic response with respect to the molar ratioR T (figure 2b). Specifically, whenR T ¼ 0, E A =E T has the maximum value 1=(1 þK a ), which becomes closer to one as A binds to DNA more tightly (i.e.K a ( 1). WhenR T ¼K b (1 þK a ), E A =E T is reduced to its half-maximal value. Thus, as the binding between the DNA-bound A and R becomes tighter (i.e.K b decreases), the transcriptional activity achieves its half-maximal value at the lowerR T (figure 2b). The sensitivity of the transcriptional activity with respect toR T can be quantified using the effective Hill coefficient Logð81Þ=Log(EC10=EC90), which is equivalent to the Hill exponent for a Hill curve [37]. The effective Hill coefficient of the transcriptional activity is one regardless of theK b andK a values (figure 2c), as expected from the Michaelis-Menten-type equation (equation (2.2)). Taken together, with the sole blocking repression, the transcriptional activity cannot sensitively respond toR T :

The combination of the sequestration-and blocking-type repressions can generate ultrasensitivity
We wondered whether the sensitivity of the transcriptional activity can be increased by incorporating an additional repression mechanism. To investigate this, we added the sequestration-type repression to the blocking model: R binds with the free A to form complex R A with a dissociation constant of K s , and thus sequesters A from DNA (figure 2d; see Methods for details). Due to the complex R A , the conservations are switched to When the binding between A and R is weak (i.e.K s ¼ K s =A T ) 1) and thus R A is negligible, the steady states of A and R can be approximated with simple A T and R T . On the other hand, when the binding is not weak, R A is not negligible and thus the approximations for the steady states of A and R become slightly complex (see Methods for details): When the binding between A and R is extremely tight (K s % 0), A and R can be approximated by the simple functions max(A T À R T , 0) and max(R T À A T , 0), respectively [21,28,30]. By substituting equation (2.3) for equation (2.1), the approximated E A =E T can be derived: The transcriptional activity described by equation (2.4) shows more sensitive responses with respect toR T compared to the blocking model as the sequestration becomes stronger (i.e.K s decreases; figure 2e). Specifically, when the sequestration is weaker than the blocking (i.e.K s .K b ), the transcriptional regulation is mainly governed by the blocking, and thus the transcriptional activity shows a hyperbolic response (figure 2e, dotted line) similar to the sole blocking-type repression (figure 2b). On the other hand, when the sequestration is stronger than the blocking (i.e.K s ,K b ; figure 2e, solid line), R is more likely to bind with the free A rather than the DNA-bound A. Thus, when there are more activators than repressors (i.e.R T , 1), the majority of R is bound to the free A, not the DNA-bound A, and thus the high level of transcriptional activity is maintained. AsR T is greater than one and thus the free R, not sequestered by the free A, is available, R can block the DNA-bound A, leading to the rapid drop in the transcriptional activity (figure 2e, solid line). This switch-like transition in the transcriptional activity generates the ultrasensitivity (figure 2e). Consistently, the effective Hill coefficient increases as the sequestration becomes stronger (i.e.K s decreases; figure 2f ).
The ultrasensitivity can be generated when the blocking and sequestration act synergistically (electronic supplementary material, figure S1). That is, when the blocking is stronger than the sequestration (K b ,K s ; electronic supplementary material, figure S1a-c), the ultrasensitivity cannot be generated, similar to the sole blocking model (figure 2c). When the blocking is too weak (K s (K b ), and thus the transcriptional regulation is mainly governed by the sequestration, the DNA-bound activator cannot be royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20210084 4 inhibited effectively via blocking. As a result, ultrasensitivity cannot be generated when the activator binds to DNA more tightly than the repressor (K a ,K s ; electronic supplementary material, figure S1d,e). Taken together, to generate ultrasensitivity, the appropriate level of blocking and stronger sequestration compared to the blocking are needed. This requires a mechanism for the repressor to have different binding affinities with the free activator and the DNA-bound activator. Furthermore, due to the requirement of stronger sequestration compared to the blocking, the condition is challenging to achieve with physiologically plausible binding affinities. Specifically, the concentration of transcriptional factors (A T ) is 2 Â 10 À9 -10 À7 M as their number is 10 4 -10 5 (i.e. 2 Â 10 À20 -10 À19 mol) and the typical mammalian cell volume is 10 À11 -10 À12 l [32,38,39]. Thus, even the extremely high affinity protein whose dissociation constant is picomolar (i.e. K s % 10 −12 M) hasK s with the range of 0:5 Â 10 À5 -10 À3 . With these physiologically plausible values ofK s , the range ofK b where the ultrasensitivity can be generated is narrow (electronic supplementary material, figure S1).

The combination of the displacement-, sequestration-and blocking-type repressions can readily generate ultrasensitivity under physiologically plausible conditions
To investigate whether the requirement of the strong sequestration can be relaxed by adding the displacementtype repression, we expanded the model where the complex R A can dissociate from DNA with a dissociation constant of K d (figure 2g; see Methods for details). Due to the displacement, E F is affected by two different reversible bindings between R A and E F as well as between A and E F unlike in the previous models. Thus, the steady state By solving these coupled equations, we can get the ratio of the steady states of E F , E A and E R , i.e. 1: where IðRÞ¼ ðK s þ sK a þ RÞ=ðK s þ sK a þ sRÞ and J(R) ¼ ðK s þ K a þ RÞ=ðK s þ sK a þ sRÞ, and s ¼ K s K d =K b K a : Note that when s ¼ 1, which is known as the detailed balance condition [40], I(R) ¼ J(R) ¼ 1 and thus the ratio becomes the same as the previous simple one. This is because under the detailed balance condition, all reversible bindings reach equilibrium, and thus the steady state equations of the species affected by multiple reversible reactions (e.g. [40]. Therefore, under the detailed balance condition, the transcriptional repression by the three types of repressions becomes equivalent to the repression by the blocking and sequestration types. When s = 1, the ratio of the steady states of E F , E A and E R are changed and thus we get 3) into A and R and normalizing the variables and parameters with A T , we can derive the approximation for E A =E T in terms of the molar ratio ðR T Þ: ð2:5Þ To investigate whether the displacement enhances the sensitivity of the transcriptional activity, we first consider the case where R binds to the free A and the DNA-bound A with the same affinity (i.e.K b ¼K s ), so the combination of the sequestration-and blocking-type repressions fails to generate the ultrasensitivity (figure 2f and electronic supplementary material, figure S1 larger asK d becomes greater thanK a (figure 2i). Furthermore, even ifK b ,K s (i.e. the sequestration is weaker than the blocking), the ultrasensitivity can be generated when the effective displacement occurs (electronic supplementary material, figure S2), unlike with the combination of blocking and sequestration (figure 2f and electronic supplementary material, figure  S1). Taken together, effective displacement can eliminate the requirement for the combination of the sequestration-and blocking-type repressions to generate the ultrasensitivity. When there is no energy expenditure, the dissociation constants have to satisfy the detailed balance condition ðs ¼K sKd =K bKa ¼ 1Þ [41]. In this case, effective displacement can occur ðK d .K a Þ under limited conditions ðK s ,K b Þ, which is challenging to achieve physiologically. On the other hand, when energy is expended to break the detailed balance condition ðs . 1Þ, effective displacement can occur without the limitation. Such energy expenditure can happen mechanistically by adenosine triphosphate hydrolysis [42].
Interestingly, when there is no energy expenditure to break the detailed balance condition, the equilibrium relations for each reversible reaction (i.e. at the steady state [40]. Thus, the transcription regulated by all three repressions becomes the same as that regulated by any two of the repressions (see electronic supplementary material for details). This allows us to easily identify the condition for ultrasensitivity generated with any two repression mechanisms by substituting the detailed balance condition (s ¼K sKd =K bKa ¼ 1) to the condition for the ultrasensitivity generated with the three repression mechanisms (electronic supplementary material, table S1). This reveals that the requirement of strong sequestration of the blocking and sequestration model, which was challenging physiologically, is switched to the effective displacement of the blocking and displacement model. Importantly, with energy expenditure, the combination of all three repressions can generate ultrasensitivity over a wider range of conditions compared to the royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20210084 combination of any two repressions (figure 2i; electronic supplementary material, table S1 and figures S2 and S3).

The transcriptional negative feedback loop with multiple repression mechanisms can generate strong rhythms
Ultrasensitivity is critical for the transcriptional NFL to generate sustained and strong oscillations [18][19][20][21]. Thus, when the transcriptional repression is regulated by the combination of the multiple repression mechanisms, the strong oscillations can be generated. To investigate this, we constructed a simple transcriptional NFL model (figure 3a), where the free activator (A) binds to the free DNA, and then promotes the transcription of the repressor mRNA (M). M is translated to the repressor protein in the cytoplasm (R c ). After translocation to the nucleus, the repressor protein (R) inhibits its own transcriptional activator (A) with the previously described repression mechanisms ( figure 2a,d,g). Thus, the transcription of M depends on the transcriptional activity E A =E T . We assumed that E A =E T rapidly reaches its quasisteady-state because the reversible bindings regulating the transcriptional activity typically occur much faster than the other processes of the transcriptional NFL (i.e. transcription, translation, translocation and degradation). Using the quasi-steady-state approximation (QSSA) and the nondimensionalization, we can obtain a simple NFL model (see electronic supplementary material for details): where E A ðR T Þ=E T is the QSSA for the transcriptional activity. Depending on the repression mechanism, we can use the steady state equations for E A =E T derived in the previous sections (i.e. equations (2.2), (2.4) and (2.5)). Note that these QSSAs are known as the 'total' QSSAs as they are determined by the molar ratio between the 'total' concentrations of the repressor and activator,R T ¼ R T =A T , which is not affected by the fast reversible bindings. Thus, the QSSAs are accurate as long as the reversible bindings are fast [43]. In this way, the NFL model (equation (2.6)) can accurately capture the dynamics of the interactions between A and R even when their levels are comparable [43]. As more repression mechanisms are added, E A ðR T Þ=E T more sensitively changes in response to the variation ofR T (figure 3b), which is critical for amplitude amplification. Thus, stronger rhythms, which have a high relative amplitude (i.e. the amplitude normalized by the peak value of the rhythm), are generated (figure 3c). Specifically, while the NFL with the sole blocking repression cannot generate rhythms (figure 3c, red dotted line), the NFL with the combination of the blocking, sequestration, and displacement can generate the strongest rhythms (figure 3c, green solid line). Such strong rhythms become weaker as the displacement becomes ineffective (i.e. K d becomes smaller thanK a ; figure 3d), or the blocking or the sequestration become weaker (i.e.K b orK s increases; figure 3e).

In the mammalian circadian clock, the disruption of synergistic multiple repressions weakens rhythms
In the transcriptional NFL of the mammalian circadian clock, the transcriptional repression occurs via the combination In particular, the combination of the blocking, sequestration, and displacement can generate the strongest rhythms. Here, A T ¼ 0:05,K a ¼ 10 À4 ,K b ¼ 10 À5 ,K s ¼ 10 À5 andK d ¼ 10 À1 are used, and all trajectories are normalized by their own maximum value to compare their relative amplitudes. (d ) As the displacement becomes ineffective (i.e.K d becomes smaller thanK a ), the relative amplitude decreases. (e) Similarly, as the blocking or sequestration becomes weaker (i.e.K b orK s increases), the relative amplitude decreases.   Figure 4. In the mammalian circadian clock, the disruption of synergistic multiple repressions weakens rhythms. (a) In the mammalian circadian clock, the transcriptional activity of CLOCK:BMAL1 is regulated by blocking-, sequestration-and displacement-type repressions. Several mutations disrupting the combination of multiple repressions have been identified. BMAL1 transactivation domain mutations such as 619X and L606A L607A decrease the binding affinity between BMAL1 and CRY1 (i.e. K b increases), weakening the blocking. CLOCKΔ19 has impaired binding with PER (i.e. K s increase), disrupting the sequestration. A CK1δ −/− mutation prevents the CK1δ-induced phosphorylation of CLOCK:BMAL1, which is essential for the effective displacement (i.e. K d decreases). Furthermore, BMAL1Δbasic has impaired binding with the Ebox (i.e. K a increases), decreasing K d /K a and thus disrupting the effective displacement. (b) Schematic diagram showing the alteration of amplitudes by change in dissociation constants K a and K d based on the predictions in figure 3d. In the mammalian circadian clock, CLOCK:BMAL1 shows higher binding affinity with the E-box compared to the PER:CRY:CLOCK:BMAL1 complex (i.e. K d =K a . 1; below the grey line) [15], which is critical for strong rhythm generation according to our model prediction. (c,d) When K a was increased by the BMAL1Δbasic mutation (c) and K d was decreased by the CK1δ −/− mutation (d ), the amplitude of PER2-LUC rhythms was reduced to 0.5 and 0.6 compared to WT mice, respectively. Adapted from [36] and [33]. (e) Schematic diagram showing the alteration of amplitudes by change in dissociation constants K s and K b based on the predictions in figure 3e. In the mammalian circadian clock, PER:CRY binds with CLOCK:BMAL1 tightly (i.e. small K s ) and CRY binds with CLOCK:BMAL1:E-box tightly (i.e. small K b ), which is crucial for strong rhythm generation according to our model prediction. ( f ) Indeed, as the dissociation constant between CRY and CLOCK:BMAL1:E-box (K b ) was increased by the BMAL1 619X mutation, the amplitude of PER2-LUC rhythms from fibroblasts in mutant mice was reduced to 0.4 compared to that in WT. The amplitude was further reduced when K b was further increased by the BMAL1 L606A L607A mutation. Adapted from [35]. (g) When the binding affinity between PER and CLOCK was decreased (i.e. K s increased) by the CLOCKΔ19 mutation, the amplitude of PER2-LUC rhythms in the SCN of mutant mice was reduced to 0.7 compared to that in WT mice. Adapted from [34]. For each mutation, all adapted PER2-LUC rhythms of WT and mutant mice were measured under the same condition. However, amplitudes among different mutations cannot be compared due to different experimental conditions. royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20210084 of blocking, sequestration and displacement (figure 4a). Specifically, CLOCK:BMAL1 binding to E-box regulatory elements in the Period (Per1 and Per2) and Cryptochrome (Cry1 and Cry2) genes activates their transcription at around circadian time (CT) 4-8. After CRY and PER are translated in the cytoplasm, they form the complex with the kinase CK1δ and enter the nucleus. The complex dissociates CLOCK:BMAL1 from the E-box and sequesters CLOCK: BMAL1 to prevent binding to the E-box at around CT12-22 (displacement-and sequestration-type repression). At around CT0-4, CRY binds to the CLOCK:BMAL1:E-box complex to block the transcriptional activity (blocking-type repression) [15][16][17]44].
In the mammalian circadian clock, because the PER:CRY complex recruits CK1δ, inducing dissociation of CLOCK: BMAL1 from the E-box [15], the binding affinity of CLOCK: BMAL1 with the E-box is higher compared to its complex with PER:CRY (i.e. K d =K a . 1). This effective displacement is critical for strong rhythm generation (figure 4b, black solid line) according to our model prediction (figure 3d). Then we can expect that, as either K a increases or K d decreases (i.e. K d =K a decreases), which deactivates the displacementtype repression, the circadian rhythms become weaker (figure 4b, orange solid line). Indeed, when K a was increased by a BMAL1 mutant lacking the basic region (BMAL1Δbasic), which is critical for the binding of BMAL1 to the E-box element ( figure 4a, top right), the amplitude of PER2-LUC rhythms from the fibroblasts of mutant mice was reduced compared to that from wild-type (WT) mice (figure 4c) [36]. Furthermore, when K d was decreased by a CK1δ −/− mutant lacking the CK1δ-induced dissociation of CLOCK:BMAL1 from the E-box ( figure 4a, bottom left), the amplitude of PER2-LUC rhythms in the suprachiasmatic nucleus (SCN) of mutant mice was also reduced compared to that in WT mice (figure 4d) [33]. Note that the amplitude reduction by the CK1δ −/− mutant could be due to other factors because CK1δ also regulates the stability and nucleus entry of PER [45].
The blocking-and sequestration-type repressions also effectively occur in the mammalian circadian clock. That is, PER:CRY binds with CLOCK:BMAL1 tightly (i.e. small K s ), and CRY binds with CLOCK:BMAL1:E-box tightly (i.e. small K b ) [46]. Such tight bindings are important for strong rhythm generation (figure 4e, black solid line) according to our model prediction (figure 3e). Thus, as either K b or K s increases, weakening the blocking-or the sequestration-type repression, the rhythms are expected to become weaker (figure 4e, orange solid lines). Indeed, when K b was increased due to the BMAL1 619X mutation reducing the binding affinity between BMAL1 and CRY1 ( figure 4a, top left), the amplitude of PER2-LUC rhythms from the fibroblasts of mutant mice was reduced to 0.4 compared to WT (figure 4f ) [35]. When K b was further increased by a BMAL1 L606A L607A mutation, the amplitude was further reduced (figure 4f ) [35]. Moreover, when K s was increased by the CLOCK mutant lacking the exon 19 region (CLOCKΔ19), which is required for the binding of PER (figure 4a, bottom right) [47], the amplitude of PER2-LUC rhythms in the SCN of mutant mice was reduced compared to that in WT mice (figure 4g) [34]. Note that such reduction of the amplitude by CLOCKΔ19 could be due to other factors such as the low transcriptional activity of CLOCKΔ19 [48] and the impaired binding with the E-box [49].

Discussion
Transcriptional repression plays a central role in precisely regulating gene expression [2]. Various mechanisms for the repression have been identified [2][3][4]. In particular, the transcriptional activators can be inhibited in various ways by repressors such as blocking, sequestration and displacement ( figure 1a). Interestingly, these repression mechanisms are used together to inhibit a transcriptional activator in many biological systems [3]. In this study, we found that multiple repression mechanisms can synergistically generate a sharply ultrasensitive transcriptional response (figure 2) and thus strong rhythms in the transcriptional NFL ( figure 3). Consistently, the mutations disrupting any of the blocking, sequestration or displacement in the transcriptional NFL of the mammalian circadian clock weaken the circadian rhythms ( figure 4). Our work identifies a benefit of using multiple repression mechanisms together, the emergence of ultrasensitive responses, which are critical for cellular regulation such as epigenetic switches, the cell cycle and circadian clocks [22].
Recently, detailed transcriptional repression mechanisms underlying various biological systems have been identified. For instance, while MDM2 was known to inhibit p53 by promoting its degradation [50], recent studies have suggested that MDM2 can also inhibit p53 through displacement and blocking [51,52]. In the Rb-E2F bistable switch, the suppressor Rb protein and the E2F family of transcription factors inhibit mutually with multiple repression mechanisms such as blocking and chromatin structure modification, which are critical to generate ultrasensitivity and thus the bistable switch of cell cycle [6,7,53]. However, such repression mechanisms have not yet been incorporated into the mathematical models [54][55][56][57]. Similarly, the recent discoveries of multiple repression mechanisms underlying biological oscillators such as the circadian clock [14][15][16][17] and the NF-kB oscillator [12,13] have not been fully incorporated even in recent mathematical models [24,[58][59][60][61][62][63]. In particular, the majority of the mathematical models for various systems have used the simple Michaelis-Menten-or Hill-type functions to describe the transcriptional repression regardless of its underlying repression mechanisms, which can distort the dynamics of the system [21,43]. Our work highlights the importance of careful modelling of the transcriptional repression depending on blocking, sequestration or displacement to accurately capture the underlying dynamics.
Interestingly, to fully use the three repression mechanisms, energy expenditure is required. Without the energy expenditure, the detailed balance condition needs to be satisfied (s ¼ K s K d =K b K a ¼ 1). Under this restriction, the transcriptions regulated by the three repression mechanisms and any two of these become equivalent (see electronic supplementary material for details). As a result, the ultrasensitivity is generated under a limited condition compared to when the detailed balance condition is broken via dissipation of energy (s . 1) (figure 2i; electronic supplementary material, table S1 and figures S1 and S2). Similarly, the limitation for generating the sensitivity of transcription under the detailed balance condition was also identified when DNA is directly regulated by its transcriptional factors [41]. Specifically, Estrada et al. found that when the energy expenditure breaks the detailed balance condition, the cooperative royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20210084 bindings of the transcriptional factors to multiple binding sites of DNA are more likely to generate ultrasensitivity.
The advantages of using multiple repression mechanisms for biological oscillators have just begun to be investigated. For instance, in the NF-kB oscillator, IkBa inhibits its own transcriptional activator NF-kB via sequestration and displacement. Wang et al. found that the displacement can enhance NF-kB oscillation by dissociating the NF-kB from decoy sites and promoting its nuclear export (i.e. facilitating the sequestration), and compensating for the heterogeneous binding affinity of NF-kB to the promoter of IkBa [64]. Furthermore, a recent study of the transcriptional NFL of the mammalian circadian clock found that the displacement of the transcriptional activator (BMAL1:CLOCK) by its repressor (PER:CRY) can facilitate the mobility of the BMAL1:CLOCK to its various target sites, pointing out the hidden role of PER:CRY [65]. While PER:CRY dissociates and sequesters CLOCK:BAML1 from E-box (i.e. sequestration and displacement type), CRY blocks the transcriptional activity of CLOCK:BMAL1 (i.e. blocking type) [15][16][17]. Because Cry1 displays a delayed expression phase compared to Per, the blocking repression occurs at the late phase, which turns out to be critical for rhythm generation [66][67][68]. It would be interesting in future work to extend the model to include multiple repressors (e.g. PER and CRY) to investigate their distinct roles.
While we focused on transcriptional repression mechanisms, other mechanisms leading to ultrasensitivity [69], and thus generating rhythms, have been identified. For instance, phosphorylation of the repressor [24,70,71] and saturated degradation of the repressor [25,32,72] can be additional sources of ultrasensitivity for strong rhythms. Furthermore, an additional transcriptional positive feedback loop has been known to enhance the robustness of rhythms [18,21,71,73] in the presence of Hill-type transcriptional repression, which can be induced by phosphorylation-based transcriptional repression [74,75]. On the other hand, when the transcription is regulated by sequestration-type repression, an additional NFL rather than the positive feedback loop can enhance the robustness of rhythms [21,28,32]. It would be important in future work to investigate the role of additional feedback loops depending on the transcriptional repression mechanisms identified in this study.
A transcriptional NFL, where a single repressor inhibits its own transcription by binding to its own promoter, is the simplest design of the synthetic genetic oscillator [76,77]. To generate the ultrasensitivity with this simple design, Stricker et al. used a repressor that forms a tetramer to bind with its own promoter [78]. Nonetheless, the degree of the ultrasensitivity was not enough for the synthetic oscillator to generate strong oscillations with high amplitude. Thus, more complex designs of synthetic oscillators have been constructed [76,77]: the modified repressilators [79,80], the combination of the negative and positive feedback loops [78,81], and the coupling of synthetic microbial consortia [82][83][84][85]. Our study proposes that a strong synthetic oscillator with a simple design (i.e. a single NFL) could be constructed by modifying the previously used repression mechanisms. That is, by using the combining blocking-, sequestration-and displacementtype repressions, although this might be challenging to implement, ultrasensitivity to achieve strong rhythms could be obtained, providing a new strategy for the design of synthetic oscillators.

The equation for the transcriptional activity regulated by the sole blocking-type repression
The transcription regulated by sole blocking-type repression (figure 2a) can be described by the following system of ordinary differential equations (ODEs) based on the mass action law: where R, A, E F , E A and E R represent the concentration of the repressor, the activator, DNA, the activator-bound DNA and the activator and repressor complex-bound DNA, respectively.
Here, k fb (k b ) and k fa (k a ) are the association (dissociation) rate constants between E A and R and between A and E F , respectively.
Note that as The steady states of the system satisfy the following equations: where K b ¼ k b =k fb and K a ¼ k a =k fa : This yields E F : E A : E R = 1: A/K a : ðR=K b ÞðA=K a Þ, and thus the steady state for E A =E T : where A and R in equation (4.3) are the steady states of the free activator and the free repressor, respectively. Equation (4.3) can be simplified if the total concentration of DNA (E T ) is much lower than the concentrations of the activator (A T ) and repressor (R T ) and thus That is, by replacing A and R in equation (4.3) with conserved A T and R T , respectively, we get the following approximation for equation (4.3): which is accurate as long as E T =A T is small (electronic supplementary material, figure S4a). This assumption is likely to hold in the mammalian circadian clock as the number of BMAL1:CLOCK in the mammalian cells is about 10 4 -10 5 [38].
On the other hand, it might not be acceptable in E. coli or S. cerevisiae cells, which contain much lower numbers of transcription factors (10 1 -10 2 ) [38].
royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20210084 9 4.2. The equation for the transcriptional activity regulated by the blocking-and sequestration-type repressions The transcription regulated by both blocking and sequestration (figure 2d ) can be described by the following ODEs: The reversible binding between R and A to form the complex (R A ) with the association rate constant k fs and the dissociation rate constant k s are added to equation (4.1). Thus, the conservation laws for the activator and the repressor are changed to Because the steady states of equation (4.5) also satisfy equation (4.2), the steady state of E A =E T in this system also satisfies equation (4.3). However, even if E T is much lower than A T and R T , equation (4.3) cannot be simplified by replacing A and R with A T and R T because Thus, we also need to use another steady state equation, AR ¼ K s R A , where K s ¼ k s =k fs , to derive the steady state of R A in terms of A T and R T . Specifically, by replacing A and R with A T À R A and R T À R A , respectively, in Then by substituting equation (4.6) to A % A T À R A and R % R T À R A , we can get the following approximate steady state for the free activator and repressor: By substituting equation (4.7) to equation (4.3), the approximate E A =E T can be derived (equation (2.4)), which is accurate as long as E T =A T is small (electronic supplementary material, figure S4b).

The equation for the transcriptional activity regulated by all the blocking-, sequestration-and displacement-type repressions
The transcription regulated by all blocking, sequestration, and displacement (figure 2g) can be described by the following ODEs: which have the same conservation laws as equation (4.5).
Because the reversible binding between R A and E F to form E R with the association rate constant k fd and the dissociation rate constant k d are added to equation (4.5), the steady states are changed to the following equations: where K d ¼ k d =k fd , g ¼ k fd =k fs , d ¼ k fb =k fa , and u ¼ k fd =k fb . If E T is much lower than A T and R T , and thus A T ¼ A þ R A þ E A þ E R % A þ R A and R T ¼ R þ R A þ E R % R þ R A , the last two equations of (4.9) can be simplified as follows: (A T À R A )ðE T À E A À E R Þ À K a E A À d(R T À R A )E A þ dK b E R % 0, (R T À R A )E A À K b E R þ uR A ðE T À E A À E R Þ À uK d E R % 0:

ð4:10Þ
Their solution yields the steady state approximation for E A =E T as : ð4:11Þ Furthermore, by using the approximation A T % A þ R A and R T % R þ R A , we can simplify the first equation of equation (4.9) as follows: Because E T is much lower than A T and R T , equation (4.12) can be further simplified to R 2 A À (A T þ R T þ K s )R A þ A T R T % 0, leading to the approximation for the steady state of R A described in equation (4.6). Then, by substituting equation (4.6) to equation (4.11), the approximated E A =E T in terms of conserved A T and R T can be derived. The approximation of E A =E T can be further simplified as follows if we assume d ¼ 1 and u ¼ 1 (i.e. the binding rates are the same): : ð4:13Þ royalsocietypublishing.org/journal/rsfs Interface Focus 12: 20210084 Each term of equation (4.13) can be transformed by using A % A T À R A , R % R T À R A , and AR % K s R A as follows: > > > > > = > > > > > ; ð4:14Þ where s ¼ K s K d =K b K a and m ¼ K b K a =K s . By substituting equation (4.14) to equation (4.13), we can derive equation (2.5) as follows: where I(R) ¼ ðK s þ sK a þ RÞ=ðK s þ sK a þ sRÞ and J(R) ¼ ðK s þ K a þ RÞ=ðK s þ sK a þ sRÞ. By substituting equation (4.7) into A and R and then normalizing the variables and parameters with A T , we can derive the approximated equation (2.5) for E A /E T in terms of the molar ratioR T . This approximation is accurate as long as E T =A T is small (electronic supplementary material, figure  S4c). Importantly, it accurately captures the cases when d and u are not one if the displacement effectively occurs (K d . K a ; electronic supplementary material, figure S5).
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