How Retroactivity Affects the Behavior of Incoherent Feed-Forward Loops
HHow Retroactivity Affects the Behavior of Incoherent Feed-ForwardLoops
Junmin Wang The Bioinformatics Graduate Program, Boston University, Boston, MA, USA
Calin Belta
The Bioinformatics Graduate Program, Boston University, Boston, MA, USA
Samuel A. Isaacson
Department of Mathematics and Statistics, Boston University, Boston, MA, USA
Abstract
An incoherent feed-forward loop (IFFL) is a network motif known for its ability to accel-erate responses and generate pulses. Though functions of IFFLs are well studied, mostprevious computational analysis of IFFLs used ordinary differential equation (ODE) mod-els where retroactivity, the effect downstream binding sites exert on the dynamics of anupstream transcription factor (TF), was not considered. It remains an open question tounderstand the behavior of IFFLs in contexts with high levels of retroactivity, e.g., in cellstransformed/transfected with high-copy plasmids, or in eukaryotic cells where a TF binds tonumerous high-affinity binding sites in addition to one or more functional target sites. Herewe study the behavior of IFFLs by simulating and comparing ODE models with different lev-els of retroactivity. We find that increasing retroactivity in an IFFL can increase, decrease, orkeep the network’s response time and pulse amplitude constant. This suggests that increas-ing retroactivity, traditionally considered as an impediment to designing robust syntheticsystems, could be exploited to improve the performance of IFFLs. We compare the behav-iors of IFFLs to negative autoregulatory loops, another sign-sensitive response-acceleratingnetwork motif, and find that increasing retroactivity in a negative autoregulated circuit canonly slow the response. The inability of a negative autoregulatory loop to flexibly handleretroactivity may have contributed to its lower abundance in eukaryotic relative to bacterialregulatory networks, a sharp contrast to the significant abundance of IFFLs in both celltypes.
Keywords:
IFFL, retroactivity, ODE, systems biology, synthetic biology
1. INTRODUCTION
Living cells sense and respond to the environment via a large variety of mechanisms.How do diverse biochemical networks, which are at the core of the process by which cells corresponding author, lead contact (email: [email protected]) Preprint submitted to Elsevier September 29, 2020 a r X i v : . [ q - b i o . M N ] S e p ense and respond to signals, yield and maintain specific functional behaviors? A widelyheld hypothesis in systems biology is that recurring network sub-structures, also knownas network motifs, play important roles therein. Network motifs capable of performingbiological functions are preserved over the course of evolution, resulting in a rate of occurrencehigher than if nodes and edges were connected at random (Alon (2007)).One of the most common three-gene network motifs in transcriptional regulatory net-works (TRN) is the incoherent feed-forward loop (IFFL), where a transcription factor (TF)activates and inhibits a downstream gene directly and indirectly (Figure 1(a)). In a pioneer-ing study guided by ordinary differential equation (ODE) models, Mangan and Alon (2003)established IFFLs as a sign-sensitive response accelerator and pulse generator (Figure 1(b)).Subsequent efforts in synthetic biology supported the findings of Mangan and Alon (2003)with compelling experimental evidence. Using the gal system in Escherichia coli ( E. coli ),Alon (2007) showed that compared to simple regulation, IFFLs can accelerate the responsetimes of a target gene. Basu et al. (2004) demonstrated the feasibility of creating syntheticpulse-generating IFFL circuits under the guidance of ODE models. In addition, IFFLs canprovide fold-change detection and buffer noise (Goentoro et al. (2009)). Osella et al. (2011),Siciliano et al. (2013), and Grigolon et al. (2016) showed that miRNA-mediated IFFLs conferprecision and stability to the target protein level despite fluctuations in upstream regulators.Although a wealth of literature has shed light on this topic, it remains an open areaof research to understand the full functional capabilities of IFFLs. In their ODE models,Mangan and Alon (2003), as well as Basu et al. (2004), made the simplifying assumptionthat changes in protein concentrations arise from first-order decay and protein productionrates regulated by upstream TFs. This assumption aligns with the traditional view of TRNsas modular systems, where the temporal dynamics of a protein depend solely upon theTFs that regulate its expression. In other words, under this assumption, the dynamics ofthe protein are not affected by the components it regulates even if the protein is also a TF.However, growing theoretical and experimental evidence suggests that TRNs are not modularbut quasi-modular. A fraction of the TF molecules are employed to form complexes withdownstream binding sites, hence becoming unavailable for additional molecular activities,such as degradation, protein-protein interaction, or regulation of other genes. Examplesof such TFs include p53 (Pariat et al. (1997)) and MyoD (Abu Hatoum et al. (1998)),both of which become resistant to degradation when bound to DNA. This phenomenon,where downstream binding sites can alter the dynamics of the upstream system, is knownas retroactivity (Del Vecchio et al. (2008)).In TRNs, retroactivity is large when the amount of TF is comparable to, or smaller than,the copy number of the downstream bindings sites, or when the affinity of such binding ishigh (Del Vecchio et al. (2008)). In synthetic biology, retroactivity is widely recognized asan essential parameter to consider in model-based circuit design (Brophy and Voigt (2014)).In the context of endogenous regulatory networks, retroactivity is seldom discussed, as thelevel of retroactivity that arises from TF binding in the genome is typically assumed tobe negligible (Jayanthi et al. (2013)). However, results from ChIP-on-chip and ChIP-seqmethods suggest that the validity of this assumption is dependent on the biological contextof the network (Kemme et al. (2016)). In particular, genome-wide studies driven by theEncyclopedia of DNA Elements (ENCODE) project have shown that in eukaryotic cells,TFs bind to not only functional sites in the cis-regulatory elements (e.g., promoters and2 ) − . . . I ndu ce r − Time . . . . C C C +C )C peak t t .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . RT = t - t .
00 0 .
25 0 .
50 0 .
75 1 . . . . . . PA = C peak - C b)c) Figure 1: (a) Graphical representations of four types of IFFL: I1-FFL, I2-FLL, I3-FFL, and I4-FFL. AnIFFL is a three-node network motif, where the input A, stimulated by an external inducer I, regulates theoutput C in two opposing directions. Arrows indicate activation, and edges with bars at the ends, inhibition.(b) Definitions of response time and pulse amplitude. Response time, abbreviated as RT, is defined as thetime needed to reach the midpoint between the pre-induction and the post-induction steady states (t -t ),whereas pulse amplitude, abbreviated as PA, is defined as the difference between the pre-induction steadystate and the peak concentration (C peak -C ). (c) The effect of accessible ND binding sites on the dynamicsof the TF. Non-functional NDs sequester some of the upstream TF, so only a fraction of the upstream TFmolecules are available to bind to functional target sites (e.g., promoters and enhancers). enhancers) but also numerous high-affinity sequence-specific binding sites that are seeminglynon-functional (Consortium (2012); Fisher et al. (2012); Li et al. (2008)) (Figure 1(c)). It hasbeen suggested that these high-affinity sequence-specific binding sites can serve as naturaldecoys (NDs), which compete with functional target sites for TF binding (Burger et al. (2010,2012); Lee and Maheshri (2012); Liu et al. (2007); Wang et al. (2016)). While the majorityof ND sites are inaccessible due to chromatin structure, CpG methylation, or competingproteins, an average TF in the human genome still has approximately 10 − accessibleND sites, which typically have greater or at least comparable binding affinity compared tosequence-specific TF binding sites (Kemme et al. (2016); Esadze et al. (2014); Kemme et al.(2015)) (Figure 1(c)). As such, in studying many eukaryotic TRNs retroactivity must betaken into account (Kemme et al. (2016)).The goal of our study is to understand how retroactivity affects response acceleration andpulsing of IFFLs. In the simplest case where an input is coupled to a downstream promoterbinding region, Del Vecchio et al. (2008) demonstrated that retroactivity increases response3imes and dampens pulse amplitude (Figure 1(d)). In the context of more complicatedtopologies, changing retroactivity can lead to more sophisticated, often undesired effects oncircuit behaviors (Sepulchre and Ventura (2013); Gyorgy and Del Vecchio (2014); Wang andBelta (2019)). This raises the question whether retroactivity is simply an impediment toovercome in designing synthetic IFFL circuits. Another natural question is the potential roleof retroactivity in motif evolution. As the levels of retroactivity differ sharply in prokaryotesand higher eukaryotes due to the number of accessible ND sites, could the behaviors ofa network motif under different levels of retroactivity have affected its abundance, as oneprogresses from bacterial TRNs to eukaryotic ones? Note, we focus on IFFLs in particularbecause synthesizing functional IFFLs has proven to be experimentally feasible (Basu et al.(2004); Bleris et al. (2011)), making our predictions experimentally testable in controlledsynthetic systems.Gyorgy and Del Vecchio (2014) developed a systematic modeling framework that accountsfor retroactivity in TRNs. Using this framework, we study IFFL networks by simulating,comparing, and mathematically analyzing ODE models with varying levels of retroactivity.Similar to previous computational studies (Shi et al. (2017); Ma et al. (2009); Castillo-Hair et al. (2015)), we performed time course simulations of IFFLs repeatedly with kineticparameters representing different regions of parameter space. We quantified the responsetime, as well as the pulse amplitude, for each parameter set (see Figure 1(b) for the definitionsof response time and pulse amplitude). Building from these simulations, we compared thedynamics of the corresponding ODE systems in order to understand how retroactivity affectsthe behavior of IFFLs. To demonstrate that our findings are parameter-independent, wecarried out mathematical proofs where model parameters can take arbitrary positive values.We find that increasing retroactivity can increase, decrease, or keep the response time andthe pulse amplitude constant in an IFFL. This suggests that in contrast to the traditionalperception of retroactivity as an impediment to circuit design (Del Vecchio et al. (2008)),increasing retroactivity could actually be harnessed to improve the performance of IFFLs.Our results predict that the introduction of synthetic decoy binding sites into a syntheticIFFL system would affect its response time and pulse amplitude, and the magnitude ofthis effect would depend on kinetic parameters (e.g., Hill coefficients) and circuit topologies(e.g. I1-FFLs). Hence, retroactivity should be considered in connection with circuit partsto optimize the behavior of IFFL circuits. Our observations of IFFLs led us to examine afew other motifs capable of sign-sensitive response-acceleration. Comparing the behavior ofIFFLs to that of negative autoregulation, we found that increasing retroactivity in a negativeautoregulated circuit can only decelerate the response. Interestingly, we observed that IFFLsare conserved in bacteria, mouse, and human networks, whereas negative autoregulatoryloops are only present in significant numbers in bacteria. The functional versatility of IFFLsat increasing levels of retroactivity, thus, may have provided IFFLs a selective advantage overnegative autoregulation in cases where decreasing or keeping the response time constant wasbeneficial.
2. RESULTS
4n this section, we describe our approach to modeling the effect of retroactivity on TRNs.A TRN can be mapped to a graph, where each node represents a gene/protein, each edgetranscriptional regulation, and the direction of an edge the direction of the regulation; acti-vation or inhibition. The time evolution of each node can be described by an ODE, wherethe time derivative represents the rate of change of the protein concentration contributed byprotein production and first-order decay. Mathematically, the rates of changes of proteins inthe network can be expressed as: d(cid:126)xdt = h ( (cid:126)x ) , (1)where h ( (cid:126)x ) = β · [(1 − γ ) H ( (cid:126)p ) + γ ] − δ x β · [(1 − γ ) H ( (cid:126)p ) + γ ] − δ x ...β n · [(1 − γ n ) H n ( (cid:126)p n ) + γ n ] − δ n x n , (2)where x i denotes the concentration of the i -th protein, and δ i , the decay rate. (cid:126)p i , theconcentration of the parent(s) of the i -th protein, is a subset of (cid:126)x . β i represents the maximalproduction rate of the i -th protein, and γ i , the basal fraction of the promoter that is active. H i is the Hill function describing the transcriptional regulation of x i by its parent(s). Ifthe i -th protein species x i has only one parent species p i , then Hill function H i ( p i ) can beexpressed as: H i ( p i ) =
11 + (cid:16) p i K i (cid:17) h i , if species p i is an inhibitor (cid:16) p i K i (cid:17) h i (cid:16) p i K i (cid:17) h i , if species p i is an activator, (3)where H i ( p i ) accounts for the fraction of the promoter that is active, K i is the dissociationconstant, and h i is the Hill coefficient. Co-regulation by multiple TFs can be modeled bysimple logic models. Unless otherwise specified, throughout this work we consider an ANDlogic, where the regulated gene is turned on only when all activators are abundant and allinhibitors are scarce (see Supplemental Information Section 1.1 for Hill functions describingco-regulation).To account for retroactivity, we adopt the framework developed by Gyorgy and DelVecchio (2014). The major assumptions needed to apply this framework are that 1) there isa separation of time scales between protein production/degradation and reversible bindingreactions between TFs and DNA, and 2) the corresponding quasi-steady state is locallyexponentially stable (Gyorgy and Del Vecchio (2014)). The first assumption is valid asprotein turnover and binding reactions typically occur on different time scales (Milo et al.(2002)). The second assumption is implicit in our use of the Hill-function-based models, andits validity is explained in Gyorgy and Del Vecchio (2014). Under these assumptions, therates of changes of protein concentrations with retroactivity considered can be described as: d(cid:126)xdt = [ I + R ( (cid:126)x )] − h ( (cid:126)x ) , (4)5here R ( (cid:126)x ), known as the retroactivity matrix (Gyorgy and Del Vecchio (2014)), can becalculated as: R ( (cid:126)x ) = (cid:40) Σ i | x i ∈ Φ V Ti R i ( (cid:126)p i ) V i if Φ (cid:54) = φ, N × N if Φ = φ. (5)Here V i is a binary matrix, containing as many rows as the length of (cid:126)p i and as many columnsas the number of nodes in the network. The element in the j -th row and k -th column of V i is 1 if the j -th parent of node i is node k , and 0 otherwise. AND logic is a special caseof independent binding, in which case R i ( (cid:126)p i ) is a diagonal matrix (see Supplemental Infor-mation Section 1.2 for calculation of R i ( (cid:126)p i )). This in turn implies that V Ti R i ( (cid:126)p i ) V i is also adiagonal matrix (Supplemental Information Section 1.4). Hence, R ( (cid:126)x ) is also diagonal. Moredetails about retroactivity, including its derivation, can be found in Gyorgy and Del Vecchio(2014). Models of IFFL networks with and without retroactivity are given in SupplementalInformation Sections 1.5 and 1.6. In this section, we describe the specific IFFL models in which we study the effect ofretroactivity, and outline our simulation protocol. IFFLs are known to be sign-sensitiveresponse accelerators and pulse generators: they accelerate or delay responses to stimulussteps only in one direction (Alon (2007); Mangan and Alon (2003)). Considering sign-sensitivity of IFFLs, we separated four IFFL motifs into two groups, one group (i.e., I1-FFL and I4-FFL) capable of response acceleration and pulse generation in response to anON step (i.e., inducer level x I changes from 0 to ∞ ) and the other (i.e., I2-FFL and I3-FFL) capable of response acceleration and pulse generation in response to an OFF step(i.e., inducer level x I changes from ∞ to 0). Here we focused on I1-FFLs and I4-FFLs,as similar analysis could be performed for I2-FFLs and I3-FFLs. We constructed non-dimensionalized ODE models for I1-FFLs and I4-FFLs (details of non-dimensionalizationcan be found in Supplemental Information Section 1.3), and simulated each model usingthe DifferentialEquations.jl package version 5.3.1 in Julia version 1.1.0 (Rackauckas andNie (2017); Bezanson et al. (2017)). We connected genes A, B, and/or C of the IFFLto additional downstream binding sites denoted by D X (X=A, B, or C). The degree ofretroactivity arising from additional downstream binding sites was allowed to vary, with theretroactivity coefficient ˜ η AD A (˜ η BD B , ˜ η CD C ) set to 0, 1.0, 10.0, and 100.0 (see SupplementalInformation Section 1.3 for definition of ˜ η XD X (X=A, B, or C)). By contrast, we assumedthat genes A, B, and C themselves are single-copy genes, and hence, retroactivity that arisesfrom binding of A, B, or C to the functional target site(s) (e.g., promoter that controls theexpression of B and C) is negligible. Note, a model without retroactivity is equivalent to amodel where ˜ η XD X equals zero.As an example, the non-dimensionalized model of an I1-FFL (Figure 1(a)) withoutretroactivity is given here: 6 ˜ x A dτ = f ˜ A = (1 − γ A ) (cid:16) x I K IA (cid:17) h IA (cid:16) x I K IA (cid:17) h IA + γ A − ˜ x A d ˜ x B dτ = f ˜ B = (1 − γ B ) (cid:16) ˜ x A ˜ K AB (cid:17) h AB (cid:16) ˜ x A ˜ K AB (cid:17) h AB + γ B − ˜ x B d ˜ x C dτ = f ˜ C = (1 − γ C ) (cid:16) ˜ x A ˜ K AC (cid:17) h AC (cid:18) (cid:16) ˜ x A ˜ K AC (cid:17) h AC (cid:19) (cid:18) (cid:16) ˜ x B ˜ K BC (cid:17) h BC (cid:19) + γ C − ˜ x C . (6)With retroactivity applied on all three nodes, the non-dimensionalized model of an I1-FFL becomes: d ˜ x A dτd ˜ x B dτd ˜ x C dτ = a b
00 0 c f ˜ A f ˜ B f ˜ C = r ADA +1 r BDB +1
00 0 r CDC +1 f ˜ A f ˜ B f ˜ C , (7)where r AD A = ˜ η AD A h AD A (cid:18) ˜ x A ˜ K AD A (cid:19) h ADA − (cid:32) (cid:18) ˜ x A ˜ K AD A (cid:19) h ADA (cid:33) − r BD B = ˜ η BD B h BD B (cid:18) ˜ x B ˜ K BD B (cid:19) h BDB − (cid:32) (cid:18) ˜ x B ˜ K BD B (cid:19) h BDB (cid:33) − (8) r CD C = ˜ η CD C h CD C (cid:18) ˜ x C ˜ K CD C (cid:19) h CDC − (cid:32) (cid:18) ˜ x C ˜ K CD C (cid:19) h CDC (cid:33) − . In Equations (6) and (7), ˜ x A , ˜ x B , and ˜ x C are the nondimensionalized concentrations ofproteins A, B, and C, whereas τ is the nondimensionalized time. f ˜ A , f ˜ B , and f ˜ C are thesums of regulated protein production and protein decay (Supplemental Information Section1.5). a , b , and c , defined as the reduction factors of d ˜ x A dτ , d ˜ x B dτ , and d ˜ x C dτ due to retroactivity,are equal to 1 if retroactivity is not considered. Note that for I4-FFLs, the only changes inEquations (6) and (7) are in the definitions of f ˜ A , f ˜ B , and f ˜ C due to the different regulatoryinteractions, i.e., r AD A , r BD B , and r CD C are still given by the same Equation (8) . Note alsothat retroactivity does not affect steady-state values of ˜ x A , ˜ x B , and ˜ x C .In terms of model simulation, we selected parameters based on values chosen by Manganand Alon (2003), exploring several orders of magnitude of parameter space. Specifically weconsidered Hill coefficients h i less than, equal to, and larger than 1 (Equation (3)). If h i is non-integer, then the underlying reaction between the promoter and the TF is likely theresultant of several mechanisms, such as chain reactions (Boekel (2009)). In this scenario,7 i , which is also the reaction order, can be considered an approximation of the detailedmechanisms (Boekel (2009)). An h i larger than, equal to, and less than 1 stands for positive,zero, and negative cooperativity, respectively. Details of the parameters can be found inSupplemental Information Section 1.3.In the absence of regulatory interactions, we assumed that only the expression of gene Ais modulated by an external inducer while genes B and C are constitutively expressed. Weinitialize our models at a steady state corresponding to a fixed inducer concentration, andsubsequently induce changes in concentrations of proteins A, B, and C via a sudden increasein the inducer’s concentration. In the case of an ON (OFF) step, the inducer level x I changesfrom 0 ( ∞ ) to ∞ (0). By integrating the ODEs until solutions reached a new steady state,we obtained one trajectory of proteins A, B, and C for each set of kinetic parameters wesampled. We begin by studying how varying levels of retroactivity on just one gene of an IFFL canalter its behaviors. That is, we allowed retroactivity on one and only one gene of the IFFL tovary, keeping retroactivity on the rest of the genes equal to zero. The response time of geneC was then calculated for each parameter set at each different level of retroactivity. Ourresults show that changing retroactivity on each node has different effects on response times,as each node of the IFFL serves a different function (Supplemental Information Sections 1.9and 1.11). While increasing ˜ η CD C expectedly slows the response time of gene C (Table S2),we observed that the response time of gene C decreases as ˜ η BD B increases, most notably for h BD B ≤ η BD B shortens the response time of gene C regardless of the values of any otherparameters for I1-FFLs (see Supplemental Information Section 1.16 for the mathematicalproof). Serving as the regulatory node in the network, gene B controls the time gap betweenthe opposing forces of regulation exerted on gene C. In response to an ON step, the expressionlevel of protein B monotonically increases. Increasing the level of retroactivity ˜ η BD B in turnslows the approach of B to steady state. Consequently, it takes protein B a longer time toeffectively repress gene C, allowing protein C to reach the half point over a shorter period oftime (Figure 2(a)). Thus, we find that increasing ˜ η BD B shortens the response time of geneC. As Table S1 indicates, the magnitude by which the response time shortens depends on h BD B as well as the IFFL topology. A detailed discussion of the underlying association canbe found in Section 2.5.When the repressor (activator) B has a strong inhibitory (activating) effect on the pro-duction of the target protein, the dynamics of C exhibit a pulse-like shape (Alon (2007)).In addition to response times, we examined how retroactivity affects pulse amplitude, whenfor a given set of parameters the IFFL generates a pulse. Increasing ˜ η CD C expectedly slowsdown the response of gene C, resulting in a lower pulse amplitude for all parameters (TableS4). In contrast, we observed and subsequently proved that increasing ˜ η BD B always increasesthe pulse amplitude (Figure S1; see Table S3 for data and Supplemental Information Section1.16 for the proof). The underlying mechanism can again be traced back to the delayedresponse of ˜ x B due to increased ˜ η BD B . While a decreased initial rate of growth of B shortensthe response time of gene C, it also causes protein B to take a longer time to effectivelyrepress gene C, allowing protein C to develop a larger response over time (Figure S1). In8 )b) Figure 2: (a) Shortened response time due to increasing ˜ η BD B in an I1-FFL. ˜ η BD B increases in the orderof top left, top right, bottom left, and bottom right. Values of the other parameters are: ˜ K AB = ˜ K AC =˜ K BC = ˜ K BD B = 0 . h AB = h AC = 1 . h BC = h BD B = 0 .
5. For comparison, the green dashed curverepresents the trajectory of ˜ x C when ˜ η BD B equals 0. The bar plot shows the response time for different˜ η BD B compared to the response time without retroactivity. (b) Shorter response time in an I1-FFL thanin a type-1 two-input circuit at different levels of ˜ η AD A . Values of the parameters are: ˜ K AB = ˜ K A B =˜ K AC = ˜ K AD A = ˜ K BC = 0 . h AB = h A B = h AC = h AD A = h BC = 1 .
0. The bar plot shows the ratio ofthe response time in an I1-FFL to the response time in a type-1 two-input circuit.
Supplemental Information Section 1.17, we extend our analysis by exploring the behavior ofan IFFL when it is embedded in a larger network and node B serves as an input to othercircuits, investigating the effect of intermodular retroactivity on IFFL behaviors. Similar to9efore, we find that increasing intermodular retroactivity on node B decreases the responsetime and increases the pulse amplitude of node C.As the input node of the IFFL, gene A regulates gene C in opposing directions. Oursimulations show that changing ˜ η AD A affects the response time and pulse amplitude of geneC in a more complicated manner than changing ˜ η BD B and ˜ η CD C . While increasing ˜ η AD A slows down the direct activation of gene C, it counteracts this delay by decelerating theactivation of gene B, thus attenuating the inhibition of C by B and allowing C a longer timeto develop a response. To demonstrate the counteracting effects, we compared the responsetime of an IFFL to that of a two-input circuit under different levels of ˜ η AD A . In a two-inputcircuit, gene C is simultaneously activated by gene A, which is induced by inducer I, andinhibited by gene A , which is induced by a separate inducer I (Figure 2(b)). To facilitatea meaningful comparison between an IFFL and a two-input circuit, we assumed that genesA and A have the same production rates upon induction, but only allowed retroactivityof gene A (not A ) to vary (see Supplemental Information Section 1.7 for the model). Wefind that because of the counteracting effects, increasing ˜ η AD A leads to a smaller increasein response time and a smaller decrease in pulse amplitude in an IFFL than in a two-inputcircuit where gene A regulates C with no feed-forward mechanism (see Tables S5, S6, S7,and S8 for data). Next, we investigated how joint increases of retroactivity on multiple nodes affect re-sponse times by letting ˜ η BD B and ˜ η CD C (˜ η AD A ) vary simultaneously. The I1-FFL model wassimulated for different values of ˜ η BD B and ˜ η CD C (˜ η AD A ) within the interval of 1.0 and 100.0.The ratio of the response time under each combination of ˜ η BD B and ˜ η CD C (˜ η AD A ) to theresponse time without retroactivity was then calculated (Figures 3(a) and (b)). We findthat in an I1-FFL, if ˜ η BD B and ˜ η CD C (˜ η AD A ) increase simultaneously, response time can beincreased, decreased, or kept constant depending on the values of ˜ η BD B and ˜ η CD C (˜ η AD A ).This is because increasing ˜ η BD B and ˜ η CD C (˜ η AD A ) affects response time in opposing direc-tions, and the resulting counteracting effects can be canceled when the values of ˜ η BD B and˜ η CD C (˜ η AD A ) satisfy a certain relationship (the solid black curves in Figures 3(a) and (b),which we call the “iso-response-time” curves).Moreover, we compared the behavior of an IFFL under increasing levels of retroactivity tothat of negative autoregulation (Figure 3(c)), another motif known for sign-sensitive responseacceleration (Rosenfeld et al. (2002)). We found that in contrast to IFFLs, increasing ˜ η AD A and/or ˜ η CD C in a negative autoregulatory circuit can only slow down the response regardlessof the values of any other parameters, as the response time of the model with retroactivityis always larger than that of the model without retroactivity (Figure 3(c); see SupplementalInformation Section 1.7 for the model and Supplemental Information Section 1.18 for theproof).Besides IFFLs and negative autoregulation, our simulations suggest that two-node neg-ative feedback loops (NFBLs) can also act as sign-sensitive response accelerators (FigureS10). Moreover, we find that if ˜ η BD B and ˜ η AD A increase simultaneously, then response timesof gene A can be increased, decreased, or kept constant depending on the values of ˜ η BD B and ˜ η AD A (Figure S10). 10 ) b) c) . . . . . log (˜ η CD C ) . . . . . l og ( ˜ η B D B ) log ( RT ˜ η BDB , ˜ η CDC /RT , ) − . . . . . . . . . . log (˜ η BD B ) . . . . . l og ( ˜ η A D A ) log ( RT ˜ η ADA , ˜ η BDB /RT , ) − . . . . . . . . . log (˜ η CD C ) . . . . . l og ( ˜ η A D A ) log ( RT ˜ η ADA , ˜ η CDC /RT , ) . . . . . . Figure 3: (a) Response times of the I1-FFL model at different levels of ˜ η BD B and ˜ η CD C compared to that ofthe model with no retroactivity. (b) Response times of the I1-FFL model at different levels of ˜ η AD A and ˜ η BD B compared to that of the model with no retroactivity. (c) Response times of the negative autoregulated circuitmodel at different levels of ˜ η AD A and ˜ η CD C compared to that of the model with no retroactivity. Valuesof parameters used for making the plots are: in (a), ˜ K AB = ˜ K AC = ˜ K CD C = 0 .
01, ˜ K BC = ˜ K BD B = 0 . h AB = h AC = h BC = h BD B = h CD C = 0 .
5; in (b), ˜ K AB = ˜ K AC = ˜ K AD A = 0 .
01, ˜ K BC = ˜ K BD B = 0 . h AB = h AC = h AD A = h BC = h BD B = 0 .
5; in (c), ˜ K AD A = ˜ K AC = ˜ K CC = ˜ K CD C = 0 . h AD A = h AC = h CC = h CD C = 0 .
5. We chose ˜ η XD X ( X = A, B, C ) to be the midpoints of the 50 subintervals that wesplit the interval [log (1 . , log (100 . η XD X ( X = A, B, C ) that were not chosen for simulation, the ratio was interpolated. The blackcurve, which we refer to as the “iso-response-time” curve, represents values of ˜ η XD X ( X = A, B, C ) at whichthe response time is the same as the response time of the model with no retroactivity. Note that in (c) thereis no “iso-response-time” curve because log (cid:16) RT ˜ ηADA, ˜ ηCDC ˜ η , (cid:17) is always larger than 1. We now examine how varying regulatory logic, e.g., I1- vs I4-FFLs and “OR” logic, canlead to different responses in the presence of retroactivity. As is demonstrated in Section2.3, increasing retroactivity ˜ η BD B accelerates the response and increases the pulse amplitudeof gene C. Our simulations also suggest that how much retroactivity affects response timeand pulse amplitude depends on the actual type of the IFFL. In response to an ON step,increasing ˜ η BD B accelerates the response times in an I1-FFL but not in an I4-FFL for h BD B ≤
1, and increases the pulse amplitude more strongly in an I1-FFL than in an I4-FFL (Figures4(a) and (b); see Tables S1 and S3 for data).The different effects of retroactivity on response time and pulse amplitude in differentIFFLs is likely an outcome of how much d ˜ x B dτ decreases in different phases of the response.To explain this argument, we take the derivative of the reduction factor b (Equation (7))with respect to ˜ x B : 11 b (˜ x B ) d ˜ x B = ˜ η BD B h BD B ˜ K h BDB − BD B · ˜ x h BDB − B (cid:32) (cid:18) ˜ x B ˜ K BD B (cid:19) h BDB (cid:33) − · (cid:34) ( h BD B − − h BD B · ˜ x h BDB B ˜ K h BDB BD B + ˜ x h BDB B (cid:35) . (9)If h BD B is a value between 0 and 1, then db (˜ x B ) d ˜ x B is always negative, indicating that b (˜ x B )is monotonically decreasing on the interval of (0 , x B transitions from a lowpre-stimulus steady state to a high post-stimulus steady state in response to an ON step.Based on monotonicity of b (˜ x B ), we know that the reduction factor is the largest when ˜ x B isclose to 0, which significantly lowers the initial value of | d ˜ x B dτ | relative to the no retroactivitycase (Figure 4(c)). Consequently, ˜ x B increases more slowly, and hence d ˜ x C dτ is significantlyincreased during the initial response phase. This results in a shortened response time andincreased pulse amplitude. On the other hand, in an I4-FFL, ˜ x B transitions from a highpre-stimulus steady state to a low post-stimulus steady state in response to an ON step(Figure 4(c)). Due to monotonicity of b (˜ x B ), the reduction factor is smallest when ˜ x B isclose to 1. This means that initially d ˜ x C dτ is minimally affected in an I4-FFL, so the effects of˜ η BD B on response time and pulse amplitude are not as strong in an I4-FFL as in an I1-FFL.If h BD B is larger than 1, then b (˜ x B ) reaches its maximum for some value of ˜ x B be-tween 0 and 1. Moreover, b (˜ x B ) monotonically increases (decreases) to the left (right) ofarg max ˜ x B b (˜ x B ). Setting db (˜ x B ) d ˜ x B equal to zero, we can obtain the following expression forarg max ˜ x B b (˜ x B ): arg max ˜ x B b (˜ x B ) = (cid:18) h BD B − h BD B + 1 (cid:19) hBDB · ˜ K BD B . (10)The qualitative behavior of the IFFL for h BD B > h BD B = 2 case.When h BD B equals 2, ˜ η BD B minimally affects response times in either I1-FFLs or I4-FFLs(Figure 4(a)). This is likely because for h BD B equal to 2 , b (˜ x B ) reaches its maximum when˜ x B reaches approximately 50% of ˜ K BD B , which happens much later than when ˜ x C reachesits half response point even in I1-FFLs (Table S10). As a result, retroactivity ˜ η BD B barelyaffects the response time when h BD B equals 2.On the other hand, an I1-FFL generally experiences a more significant change in pulseamplitude than an I4-FFL as ˜ η BD B increases (Figure 4(b)). In order to generate a pulse, ˜ x B often needs to get larger (smaller) than ˜ K BC so that it can effectively inhibit C in an I1-FFL(I4-FFL) (Table S11), which happens after b (˜ x B ) reaches its maximum. In response to an ONstep, ˜ x B transitions from a low state to a high state in an I1-FFL, so according to Equation(9), the reduction factor is the largest in the initial response phase before ˜ x B becomes largerelative to ˜ K BC , greatly lowering the initial value of | d ˜ x B dτ | (Figure S2). In contrast, in anI4-FFL, because ˜ x B transitions from a high state to a low state, the reduction factor is thelargest when ˜ x B becomes small relative to ˜ K BC somewhere in the return phase (Equation(9)) (Figure S2). As a result, ˜ η BD B affects pulse amplitude more strongly in an I1-FFL thanin an I4-FFL for h BD B larger than 1, similar to the h BD B ≤ .01 0.03 0.1 0.3 1.010 ̃ η B D B I - FF L I - FF L ̃K BD B h BD B = 0.5 h BD B = 1.0 h BD B = 2.0 a) ̃ η B D B I - FF L I - FF L ̃K BD B h BD B = 0.5 h BD B = 1.0 h BD B = 2.0 b)c) Figure 4: (a) Relative response time of I1-FFL and I4-FFL models with different values of ˜ K BD B . Here,relative response time is defined as the ratio of the response time of the model to the response time of themodel without retroactivity. Values of the parameters are: ˜ K AB = ˜ K AC = 0 . h AB = h AC = 1 .
0. (b) Pulseamplitude of I1-FFL and I4-FFL models with different values of ˜ K BD B . Values of the parameters are thesame as in (a). Note that pulse amplitudes of I1-FFLs and I4-FFLs should not be compared column-wise, asI1-FFLs generate larger pulses with larger ˜ K BD B , while I4-FFLs generate larger pulses with smaller ˜ K BD B .(c) The effect of retroactivity ˜ η BD B on response times is more pronounced in an I1-FFL (top row) than in anI4-FFL (bottom row) for h BD B ≤
1. For comparison, the green dashed curves represent the trajectories of ˜ x C when ˜ η BD B equals 0. Values of the parameters are: ˜ K AB = ˜ K AC = ˜ K BC = ˜ K BD B = 0 . h AB = h AC = 1 . h BC = h BD B = 0 . x I changes from ∞ to 0). OR logicis another special case of independent binding, where either the presence of an activator orthe absence of an inhibitor is sufficient to turn on the expression of the regulated gene (seeSupplemental Information Section 1.14 for model details). Though response time is moresensitive to changes in ˜ η BD B in an I1-FFL than in an I4-FFL under the assumption of ANDlogic, the reverse becomes true under the assumption of OR logic: in response to an OFFstep, increasing ˜ η BD B decreases the response time more strongly in an I4-FFL than in anI1-FFL (Figure S3). This is because in response to an OFF step, ˜ x B transitions from ahigh pre-stimulus steady state to a low post-stimulus steady state in an I1-FFL whereas ˜ x B transitions from a low pre-stimulus steady state to a high post-stimulus steady state in anI4-FFL (Figure S3; see Supplemental Information Section 1.14 for the data). To demonstrate the robustness of our findings, we performed extensive simulations on anI1-FFL model, where Hill coefficients, binding affinity, and decay rates were all allowed tovary. Similar to our earlier simulations, we set ˜ η BD B equal to 0, 1.0, 10.0, and 100.0, and h BC equal to 0.5, 1.0, and 2.0, to represent different levels of retroactivity and cooperativity. Therest of the parameters were sampled from their corresponding ranges via Latin HypercubeSampling (details of parameter sampling can be found in Supplemental Information Section1.19). The assumption of isometry, where TFs bind to the functional target site and non-functional decoy sites with equal affinity and cooperativity (i.e., h BC = h BD B and ˜ K BC =˜ K BD B ) was also relaxed. Instead, we assume that h BC ( h BD B ) and ˜ K BC ( ˜ K BD B ) may beunequal but correlated, as the target sites and decoy sites we considered here have the samebinding motifs. To preserve correlation, we sampled h BD B and ˜ K BD B from the intervals(1 − c ) h BC ≤ h BD B ≤ (1 + c ) h BC and (1 − c ) ˜ K BC ≤ ˜ K BD B ≤ (1 + c ) ˜ K BC , where flexibilitycoefficient c equals 0, 0.2, or 0.5. The process of ODE simulation was repeated for 10000sets of parameters.After the simulation was completed, we separated the trajectories by h AB and h AC evenlyinto 10 x 10 voxels. Within each voxel, we calculated the median relative response time(Figure 5) as well as the percent of trajectories achieving relative response time less than90% of the model in the absence of retroactivity (i.e., 10% response acceleration) (FigureS5). The results suggest that parameter isometry, which we had assumed earlier (e.g., h AB = h AC , ˜ K AB = ˜ K AC , h BC = h BD B , ˜ K BC = ˜ K BD B , δ A = δ B = δ C ), is not essential toour conclusions. Moreover, the model exhibits significant response acceleration in a largeregion of parameter space (Figure 5 and Figure S5), including regions where h AB > h AC >
1. Simulation results assuming flexibility coefficient c equal to 0 and 0.5 exhibitsimilar patterns (Figures S6, S7, S8, and S9) and further corroborate the mathematicalproof in Supplemental Information Section 1.16.
3. DISCUSSION
In this work, we studied how retroactivity affects the behavior of IFFLs via simulationand mathematical analysis. Our findings can be summarized as follows. First, in IFFLs,increasing retroactivity of the input node A, ˜ η AD A , induces counteracting effects on response14 .50.81.11.41.72.0 h BC = 0.5 h BC = 1.0 ̃ η B D B = . h BC = 2.0 h A B ̃ η B D B = . AC ̃ η B D B = . Figure 5: Response acceleration can occur over a wide set of parameters when node B regulates node Cthrough negative or neutral cooperativity. Median relative response time of I1-FFLs as model parametersare systematically varied, with random sampling of h BD B and ˜ K BD B within the intervals 0 . h BC ≤ h BD B ≤ . h BC and 0 . K BC ≤ ˜ K BD B ≤ . K BC . The trajectories are separated evenly by h AB and h AC into100 voxels, the color of which represents the median relative response time of the 100 simulated trajectorieswithin that bin, where for each trajectory the remaining parameters were chosen via random Latin HypercubeSampling. Here, relative response time is defined as the ratio of the response time of the model to the responsetime of the model without retroactivity. See Supplemental Information Section 1.19 for details of parametersampling. time and pulse amplitude, slowing both the direct activation and the indirect inhibitionof node C. Second, increasing retroactivity of the regulatory node B, ˜ η BD B , can shortenresponse times and increase pulse amplitudes, particularly in an I1-FFL with AND logic andan I4-FFL with OR logic. As a result, compared to negative autoregulation, IFFLs exhibita larger variety of functional capabilities at high levels of retroactivity. While mathematicalproofs in Supplemental Information Sections 1.16 and 1.17 demonstrate that our secondfinding is parameter-independent, the simulations systematically exploring parameter spacein Section 2.6 show that the magnitude by which retroactivity affects response times in IFFLsis significant in large regions of parameter space.
15n synthetic biology, our work lends novel insights into designing gene circuits. Mostprior studies have focused on the disruptive effects of retroactivity on the intended behaviorof circuits, e.g., shrinking the bistable region of a toggle switch (Gyorgy and Del Vecchio(2014); Gardner et al. (2000)). In contrast, here we showed that increasing retroactivitymay be used as a strategy to improve the behavior of IFFLs, i.e., creating synthetic IFFLswith shorter response times and larger pulse amplitudes while maintaining the same steady-state behaviors. One approach to changing retroactivity in synthetic systems is to mimicNDs by adding synthetic decoy sites, i.e., recombined bacterial plasmids that contain high-affinity sequence-specific binding sites. Biologically, the number of synthetic decoys can beadjusted by changing the transformation/transfection protocol, including the plasmid dose,the transformation/transfection reagent, and/or the method of transformation/transfection.Via a mechanism similar to NDs, synthetic decoys can affect the behavior of synthetic circuitsby sequestering TFs.Our work also suggests that topology alone does not constitute the entire solution tocircuit design. As shown by Figures 4 and 5, retroactivity affects the behaviors of I1-FFLsmuch more strongly for h BD B ≤ h BD B > From an evolutionary perspective, we hypothesize that the behaviors of IFFLs and neg-ative autoregulated circuits under increasing levels of retroactivity may have shaped therelative abundance of sign-sensitive response-accelerating motifs in different organisms. Us-ing published databases of
E. coli , mouse, and human TRNs (
E. coli : RegulonDB v10developed by Santos-Zavaleta et al. (2018); mouse and human: TTRUST v2 developed byHan et al. (2018)), we compared the number of times an IFFL is observed in the TRN ofeach organism to the number of times an IFFL is expected in the corresponding randomized16 igure 6: Tuning retroactivity is predicted by models to increase the response time of a synthetic IFFLconstruct (Davidsohn et al. (2015); Wang et al. (2018, 2019)). The level of retroactivity may be adjustedvia the delivery of plasmids containing synthetic decoy binding sites. See Supplemental Information Section1.20 for details.
Erdos-Renyi (ER) networks. We observed a total of 1258, 470, and 1171 IFFLs in
E. coli ,mouse, and human TRNs, whereas only 11, 5, and 7 would be expected, respectively if TF-gene interactions were completely randomized (see Supplemental Information Section 1.22for details). The number of times an IFFL is observed versus expected suggests that IFFLsare conserved in both prokaryotic and eukaryotic organisms. In contrast, the occurrence ofnegative autoregulation differs drastically between prokaryotes and higher eukaryotes. Inagreement with Stewart et al. (2013), we found that while almost half of all repressors in
E.coli are negatively self-regulated, approximately only one percent of repressors in the mouseand the human genomes are negatively autoregulated (Supplemental Information Section1.22).From a general standpoint, higher eukaryotes have much larger genomes and non-codinggenomes than prokaryotes. A direct consequence is that while an average
E. coli
TF has 3- 25 binding sites in the genome (Gao et al. (2018)), an average human TF, as is mentionedin Section 1, has approximately 10 − accessible ND sites. The degree of retroactivitythat arises from accessible ND sites is, thus, expected to be substantially higher in highereukaryotes such as mouse and human TRNs than in bacteria TRNs.Due to technical challenges, studies quantifying the cooperativity of TF-DNA binding aremuch less common in natural systems than in synthetic systems. Nevertheless, a ChIP-Seq-based study led by Ghosh et al. (2019) examining the DNA-binding of eight common TFs(i.e., Oct4, Nanog, CTCF, IRF2, FoxA1, NFAT, IRF1, and RelA) suggested that negativecooperativity and non-cooperativity may be a prevalent phenomenon of TF-DNA binding inmammalian cells (Supplementary Figure 5b of Ghosh et al. (2019)). As observed by Ghoshet al. (2019), the ChIP-seq signal strength of most of these TFs stays relatively constantor decreases as the number of binding sites increases, which indicates non- and negativecooperativity. Querying these TFs against the TTRUST v2 database (Han et al. (2018)),we found that three TFs, namely, RelA, IRF1, and CTCF, act as intermediate regulators(node B) in 101 IFFLs in human TRNs. RelA and IRF1, the latter of which exhibits clearnegative cooperativity (Supplementary Figure 5b of Ghosh et al. (2019)), serve as the inputnode (node A) and regulatory node (node B) of IFFLs regulating BCL2, CCNB1, CDK4,CDKN1A, and FOXP3, as a part of the interferon pathway (Han et al. (2018); Kochupurakkalet al. (2015)). As such, they are physical examples of IFFLs where protein B binds to DNAwith negative cooperativity. 17s is shown in Section 2.4 and Supplemental Information Section 1.18, higher retroactiv-ity in a negative autoregulatory loop results in a longer response time. This indicates thata negative autoregulatory loop achieves its minimum response time under the condition ofzero retroactivity. In contrast, as is shown in Section in 2.3 and Supplemental InformationSection 1.16, an IFFL with retroactivity on the regulatory node B can achieve response timesshorter than that of an IFFL with zero retroactivity, especially if protein B binds to DNAwith negative cooperativity or non-cooperativity. One can speculate that network motifsthat exhibit a larger diversity of functional capabilities under a high level of retroactivityare more likely to be conserved in higher eukaryotes. This is because the desired outcomeof increasing retroactivity, i.e., whether the response time should increase, decrease, or stayconstant, depends on the actual biological context, and a network motif that exhibits a largerdiversity of functions is more likely to meet the expectation of the context. This suggeststhat IFFLs could enable organisms to better adapt to a large number of accessible ND sitesduring evolution than negative autoregulation in cases where a response time shorter thanthat of the circuit under zero retroactivity is desired. Therefore, IFFLs might confer uponthe organism a selective advantage compared to negative autoregulation at high levels ofretroactivity.It is interesting to note that, in contrast to the decreased abundance of autoregulatoryloops, we have observed an increased abundance of two-node NFBLs in higher eukaryoticTRNs compared to bacterial TRNs (Supplemental Information Section 1.22). This maybe because, similar to IFFLs, two-node NFBLs also exhibit a larger diversity of functionalcapabilities than negative autoregulatory loops under high levels of retroactivity. Our work can be generalized and extended in several directions. First, it would beinteresting to explore how the connection of IFFLs to additional network motifs, such asNFBLs, affects the ability of IFFLs to accelerate responses. In a modeling study, Joanitoet al. (2018) proposed that in
Arabidopsis thaliana , the CCA1/LHY-PRR9/7(PRR5/TOC1)-CCA1/LHY IFFL circuit serves to break the bistability generated by the double NBFLsbetween CCA1/LHY and PRR5/TOC1. Compared to a plain NFBL, the IFFL-NFBL com-bination allows cells to switch between the two states more rapidly (Joanito et al. (2018)).In addition, Reeves (2019) and Ma et al. (2009) demonstrated that conjoining an NFBL toan IFFL can also increase the robustness of IFFL-mediated adaptation. That is, perfector near-perfect adaptation can be achieved over a wider region of parameter space in anIFFL-NFBL combination than in either motif alone. Studying the effect of retroactivitythat arises from the interconnection of an IFFL to an NFBL will serve to inform the designof a robust IFFL-NFBL synthetic system.More generally, the modeling framework we apply here is based on ODEs. Other ap-proaches to studying retroactivity include stochastic gene expression models, which taketranscriptional bursting into consideration. Via stochastic simulation, Kim and Sauro (2011)found that retroactivity can dampen fluctuations and lengthen correlations in the outputsignal noise when the output of a network is connected to a downstream module. It willbe interesting to study whether retroactivity can further enhance the ability of IFFLs toattenuate the stochastic variation in gene expression.18 imitations of the Study
This study presents a minimal model of IFFL circuits with and without retroactivity. Incases where two transcription factors bind to the same promoter, the model excludes bindingtypes other than AND and OR logics, such as the competitive logic.
Methods
All methods can be found in the “Transparent Methods” section of the SupplementalInformation.
Data and Code Availability
A Julia script for implementing and solving the ODEs that model IFFLs, type-1 two-inputcircuits, and negative autoregulated circuits can be found online at https://github.com/wang-junmin/IFFL.
SUPPLEMENTAL INFORMATION
Supplemental Information including proof of diagonality of retroactivity matrices, ODEmodels for IFFLs and other response-acceleration motifs, response time and pulse amplitudeof IFFLs and two-input circuits at different levels of retroactivity, response time of IFFLswith OR logic, proofs of the effects of retroactivity on response time and pulse amplitude,simulation results based on systematically exploring parameter space, an example of a sim-ulated synthetic IFFL, two-node NFBLs, and motif abundance can be found online togetherwith this article.
ACKNOWLEDGEMENTS
The authors thank Prof. Domitilla Del Vecchio, Prof. Daniel Segr`e, and Brian Teaguefor helpful discussions and constructive feedbacks. SAI was supported by National ScienceFoundation awards DMS-1255408 and DMS-1902854.
AUTHOR CONTRIBUTIONS
J.W. conceptualized the ideas, took the lead in writing the manuscript, and supervisedthis work. J.W., C.B., and S.A.I. developed the computational models. J.W. performedthe simulations, interpreted the results, and wrote the mathematical proofs with input fromS.A.I, who provided critical feedback and co-interpreted the results. All author contributedto the writing of the manuscript.
DECLARATION OF INTERESTS
The authors declare no conflict of interest.19 eferences
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Hill functions are commonly used to model transcriptional regulation in ODE models. Here, we con-sider the case where species x i is regulated by multiple TFs. Let m i be the number of parents of node i and M i be the collection of all nonempty subsets of ~ p i , i.e., {1,2, ..., m i }. Then under the assumptionof independent binding, H i ( ~ p i ) can be expressed as: H i ( ~ p i ) = P X ∈ M i π i , X Q j ∈ X ‡ p i j K i j · h i j + P X ∈ M i Q j ∈ X ‡ p i j K i j · h i j , (1)where p i j , h i j , and K i j are counterparts of p i , h i , K i for the j -th parent of node i . Similar to Gyorgyand Del Vecchio (2014), here we assume that no parents of the same node are identical. X corre-sponds to each complex formed by a different combination of TFs, and π i , X denotes the normalizedproduction rate of species x i due to the corresponding complex.Under the assumption of AND logic, the regulated gene is turned on only when all the correspond-ing activators aggregate and bind to the promoter. Let M i , A = { j | j ∈ {1,2, ..., m i } and p i j is an activator}denote the complex formed by all the activators. Then π i , X = (
1, if X= M i , A ,0, otherwise. (2) Under the assumption of AND logic, R i ( ~ p i ) is a diagonal matrix, and the k -th entry on the diagonal r ik is (Gyorgy and Del Vecchio (2014)): r ik = η i h ik p h ik − ik K h ik ik ˆ + (cid:181) p ik K ik ¶ h ik ! − , (3)where η i stands for the number of downstream binding sites (DNA copy number) of node i . p ik , h ik ,and K ik are the protein concentration, the Hill coefficient, and the dissociation coefficient of the k -thparent of node i . In order to reduce the dimensions of parameter space, we non-dimensionalized our models viamethods shown in Cao et al. (2016). We rescaled the model parameters via the following equations:˜ x A = x A δ A β A ˜ x B = x B δ B β B ˜ x C = x C δ C β C ˜ K AB = K AB δ A β A ˜ K AC = K AC δ A β A ˜ K BC = K BC δ B β B ˜ K AD A = K AD A δ A β A ˜ K BD B = K BD B δ B β B ˜ K CD C = K CD C δ C β C ˜ η AD A = η A K AD A ˜ η BD B = η B K BD B ˜ η CD C = η C K CD C (4)For simplicity, we assumed proteins A, B, and C have equal decay rates, i.e., δ A = δ B = δ C . To non-dimensionalize time, we rescaled t against the mean lifetime (equal to the reciprocal of the decayrate): τ = t δ = t · δ . (5)2on-dimensionlized models of IFFLs with and without retroactivity, are provided in SupplementalInformation Section 1.6. To simplify our analysis, we assume the following for all simulations carriedout in this work except Section 2.6 and Supplemental Information Section 1.19: each protein binds toits downstream binding sites, including both the functional target site and accessible ND sites, withequal affinity and equal cooperativity, i.e., ˜ K AB = ˜ K AC = ˜ K AD A , ˜ K BC = ˜ K BD B , h AB = h AC = h AD A , and h BC = h BD B . To ensure sufficient coverage of parameter space, we sampled the kinetic parameters˜ K X D X (X=A, B, C) spanning two orders of magnitude: ˜ K X D X ∈ {0.01, 0.03, 0.1, 0.3,1.0} and includedboth positive and negative cooperative binding, sampling Hill coefficients h X D X ( X = A , B , C ) at 0.5,1,0, and 2.0 (Mangan and Alon (2003)). The retroactivity coefficient ˜ η is the total concentration ofthe accessible ND sites for a given TF divided by the corresponding dissociation constant (Wangand Belta (2019)). Without loss of generality, the basal fraction of the promoter that is active, γ X ( X = A , B , C ) is assumed to be 10 − . 3 .4 Proof of Diagonality Here, we prove that V Ti R i ( ~ p i ) V i is a diagonal matrix. Let V Ti be an n × m matrix and a k j denote the( k , j )-th entry of V Ti . Recall that each row of V i has only one non-zero entry by definition. Similarly,let R i ( ~ p i ) be an m × m diagonal matrix and x k j denote the ( k , j )-th entry of R i ( ~ p i ). This impliesthat the product V Ti R i ( ~ p i ) is an n × m matrix. The ( k ∗ , j ∗ )-th entry of the product V Ti R i ( ~ p i ) can beexpressed as: m X u = a k ∗ u x u j ∗ = a k ∗ j ∗ x j ∗ j ∗ ,as R i ( ~ p i ) is diagonal. The ( ˆ k , ˆ j )-th entry of the product V Ti R i ( ~ p i ) V i can be expressed as: n X v = ¡ a ˆ kv x vv ¢ a ˆ j v = n X v = ‡ a ˆ kv a ˆ j v · x vv .Here a ˆ kv a ˆ j v = k ˆ j as otherwise there would be two non-zero entries in the same row of V i ,which contradicts the definition of V i . Hence, the ( ˆ k , ˆ j )-th entry of the product V Ti R i ( ~ p i ) V i is alwayszero if ˆ k ˆ j . In other words, V Ti R i ( ~ p i ) V i is a diagonal matrix. Without retroactivity, the ODE model for the I1-FFL is given as: d x A d t = f A = β A ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − δ A x A d x B d t = f B = β B ¡ − γ B ¢ ‡ x A K AB · h AB + ‡ x A K AB · h AB + γ B − δ B x B d x C d t = f C = β C ¡ − γ C ¢ ‡ x A K AC · h AC (cid:181) + ‡ x A K AC · h AC ¶ (cid:181) + ‡ x B K BC · h BC ¶ + γ C − δ C x C .I2-FFL: d x A d t = f A = β A ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − δ A x A d x B d t = f B = β B − γ B + ‡ x A K AB · h AB + γ B − δ B x B d x C d t = f C = β C − γ C (cid:181) + ‡ x A K AC · h AC ¶ (cid:181) + ‡ x B K BC · h BC ¶ + γ C − δ C x C .43-FFL: d x A d t = f A = β A ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − δ A x A d x B d t = f B = β B ¡ − γ B ¢ ‡ x A K AB · h AB + ‡ x A K AB · h AB + γ B − δ B x B d x C d t = f C = β C ¡ − γ C ¢ ‡ x B K BC · h BC (cid:181) + ‡ x A K AC · h AC ¶ (cid:181) + ‡ x B K BC · h BC ¶ + γ C − δ C x C .I4-FFL: d x A d t = f A = β A ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − δ A x A d x B d t = f B = β B − γ B + ‡ x A K AB · h AB + γ B − δ B x B d x C d t = f C = β C ¡ − γ C ¢ ‡ x A K AC · h AC ‡ x B K BC · h BC (cid:181) + ‡ x A K AC · h AC ¶ (cid:181) + ‡ x B K BC · h BC ¶ + γ C − δ C x C .When only retroactivity on A is considered, the retroactivity matrix R ( ~ x ), which is the same for allfour IFFLs, is calculated as (Gyorgy and Del Vecchio (2014)): R ( ~ x ) = η D A h ADA x AhADA − K hADAADA (cid:181) + ‡ x A K ADA · h ADA ¶ − .When only retroactivity on B is considered, the retroactivity matrix R ( ~ x ), which is the same for allfour IFFLs, is calculated as (Gyorgy and Del Vecchio (2014)): R ( ~ x ) = η D B h BDB x BhBDB − K hBDBBDB (cid:181) + ‡ x B K BDB · h BDB ¶ −
00 0 0 .When only retroactivity on C is considered, the retroactivity matrix R ( ~ x ), which is the same for allfour IFFLs, is calculated as (Gyorgy and Del Vecchio (2014)): R ( ~ x ) = η D C h CDC x C hCDC − K hCDCCDC (cid:181) + ‡ x C K CDC · h CDC ¶ − .5 .6 Non-Dimensionalized ODE Models for IFFL Without retroactivity, the non-dimensionalized ODE model for the I1-FFL is given as: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x B d τ = f ˜ B = ¡ − γ B ¢ ‡ ˜ x A ˜ K AB · h AB + ‡ ˜ x A ˜ K AB · h AB + γ B − ˜ x B d ˜ x C d τ = f ˜ C = ¡ − γ C ¢ ‡ ˜ x A ˜ K AC · h AC (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x B ˜ K BC · h BC ¶ + γ C − ˜ x C .I2-FFL: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x B d τ = f ˜ B = − γ B + ‡ ˜ x A ˜ K AB · h AB + γ B − ˜ x B d ˜ x C d τ = f ˜ C = − γ C (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x B ˜ K BC · h BC ¶ + γ C − ˜ x C .I3-FFL: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x B d τ = f ˜ B = ¡ − γ B ¢ ‡ ˜ x A ˜ K AB · h AB + ‡ ˜ x A ˜ K AB · h AB + γ B − ˜ x B d ˜ x C d τ = f ˜ C = ¡ − γ C ¢ ‡ ˜ x B ˜ K BC · h BC (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x B ˜ K BC · h BC ¶ + γ C − ˜ x C .I4-FFL: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x B d τ = f ˜ B = − γ B + ‡ ˜ x A ˜ K AB · h AB + γ B − ˜ x B d ˜ x C d τ = f ˜ C = ¡ − γ C ¢ ‡ ˜ x A ˜ K AC · h AC ‡ ˜ x B ˜ K BC · h BC (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x B ˜ K BC · h BC ¶ + γ C − ˜ x C .6 ( ~ ˜ x ) with only retroactivity on A can be written as (Gyorgy and Del Vecchio (2014)): R ( ~ ˜ x ) = ˜ η AD A h AD A ‡ ˜ x A ˜ K ADA · h ADA − (cid:181) + ‡ ˜ x A ˜ K ADA · h ADA ¶ − . R ( ~ ˜ x ) with only retroactivity on B can be written as (Gyorgy and Del Vecchio (2014)): R ( ~ ˜ x ) = η BD B h BD B ‡ ˜ x B ˜ K BDB · h BDB − (cid:181) + ‡ ˜ x B ˜ K BDB · h BDB ¶ −
00 0 0 . R ( ~ ˜ x ) with only retroactivity on C can be written as (Gyorgy and Del Vecchio (2014)): R ( ~ ˜ x ) = η CD C h CD C (cid:181) ˜ x C ˜ K CDC ¶ h CDC − ˆ + (cid:181) ˜ x C ˜ K CDC ¶ h CDC ! − .7 .7 Non-Dimensionalized ODE Models for Other Sign-SensitiveResponse-Acceleration Motifs Without retroactivity, the non-dimensionalized ODE model for a type-1 two-input circuit is given as: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x B d τ = f ˜ B = ¡ − γ B ¢ ‡ ˜ x A ˜ K A B · h A B + ‡ ˜ x A ˜ K A B · h A B + γ B − ˜ x B d ˜ x C d τ = f ˜ C = ¡ − γ C ¢ ‡ ˜ x A ˜ K AC · h AC (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x B ˜ K BC · h BC ¶ + γ C − ˜ x C .The non-dimensionalized ODE model for a type-4 two-input circuit is given as: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x B d τ = f ˜ B = ¡ − γ B ¢ + ‡ ˜ x A ˜ K A B · h A B + γ B − ˜ x B d ˜ x C d τ = f ˜ C = ¡ − γ C ¢ ‡ ˜ x A ˜ K AC · h AC ‡ ˜ x B ˜ K BC · h BC (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x B ˜ K BC · h BC ¶ + γ C − ˜ x C .The non-dimensionalized ODE model for a negative autoregulated circuit is given as: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x C d τ = f ˜ C = ¡ − γ C ¢ ‡ ˜ x A ˜ K AC · h AC (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x C ˜ K CC · h CC ¶ + γ C − ˜ x C .When only retroactivity on A is considered, the retroactivity matrix R ( ~ ˜ x ), which is the same for bothtype-1 and type-4 two-input circuits, is given as: R ( ~ ˜ x ) = ˜ η AD A h AD A ‡ ˜ x A ˜ K ADA · h ADA − (cid:181) + ‡ ˜ x A ˜ K ADA · h ADA ¶ − .8hen only retroactivity on A is considered, the retroactivity matrix R ( ~ ˜ x ) for a negative autoregu-lated circuit is given as: R ( ~ ˜ x ) = ˜ η AD A h AD A ‡ ˜ x A ˜ K ADA · h ADA − (cid:181) + ‡ ˜ x A ˜ K ADA · h ADA ¶ − .When only retroactivity on C is considered, the retroactivity matrix R ( ~ ˜ x ) for a negative autoregu-lated circuit is given as: R ( ~ ˜ x ) = η CD C h CD C (cid:181) ˜ x C ˜ K CDC ¶ h CDC − ˆ + (cid:181) ˜ x C ˜ K CDC ¶ h CDC ! − .9 .8 Pulsing Behavior of I1-FFL at Different Levels of ˜ η BD B Figure S1:
Increasing pulse amplitude due to increasing ˜ η BD B in an I1-FFL. ˜ η BD B increases in the order of topleft, top right, bottom left, and bottom right. Values of the other parameters are: ˜ K AB = ˜ K AC = ˜ K BC = ˜ K BD B = h AB = h AC = h BC = h BD B = x C when ˜ η BD B equals 0. .9 Response Time of IFFLs at Different Levels of ˜ η BD B and ˜ η CD C I1-FFL I4-FFL h BD B = K BD B ˜ η BD B h BD B = K BD B ˜ η BD B h BD B = K BD B ˜ η BD B Table S1:
Response time of gene C in IFFL models with different values of ˜ K BD B , h BD B , and ˜ η BD B (valuesrounded to two decimal places). Values of the other parameters are: ˜ K AB = ˜ K AC = h AB = h AC = I1-FFL I4-FFL h CD C = K CD C ˜ η CD C h CD C = K CD C ˜ η CD C h CD C = K CD C ˜ η CD C Table S2:
Response time of gene C in IFFL models with different values of ˜ K CD C , h CD C , and ˜ η CD C (valuesrounded to two decimal places). Values of the other parameters are: ˜ K AB = ˜ K AC = ˜ K BC = h AB = h AC = h BC = .10 Pulse Amplitude of IFFLs at Different Levels of ˜ η BD B and ˜ η CD C Here, a trajectory is considered to contain a pulse if the the trajectory maximum is larger than thepre-induction and post-induction steady states. A trajectory not satisfying this criteria is labeled as"NA".
I1-FFL I4-FFL h = K BD B ˜ η BD B h = K BD B ˜ η BD B h = K BD B ˜ η BD B Table S3:
Pulse amplitude of gene C in models with different values of ˜ K BD B , h BD B , and ˜ η BD B (values roundedto two decimal places). Values of the other parameters are: ˜ K AB = ˜ K AC = h AB = h AC = I1-FFL I4-FFL h = K CD C ˜ η CD C h = K CD C ˜ η CD C h = K CD C ˜ η CD C Table S4:
Pulse amplitude of gene C in models with different values of ˜ K CD C , h CD C , and ˜ η CD C (values roundedto two decimal places). Values of the other parameters are: ˜ K AB = ˜ K AC = ˜ K BC = h AB = h AC = h BC = .11 Response Time of IFFLs and Two-Input Circuits at Different Levels of ˜ η AD A I1-FFL Type-1 Two-Input Circuit h AD A = K AD A ˜ η AD A h AD A = K AD A ˜ η AD A h AD A = K AD A ˜ η AD A Table S5:
Response time of gene C in I1-FFLs and type-1 two-input circuits with different values of ˜ K AD A , h AD A ,and ˜ η AD A (values rounded to two decimal places). ˜ K BC = h BC = I4-FFL Type-4 Two-Input Circuit h AD A = K AD A ˜ η AD A h AD A = K AD A ˜ η AD A h AD A = K AD A ˜ η AD A Table S6:
Response time of gene C in I4-FFLs and type-4 two-input circuits with different values of ˜ K AD A , h AD A ,and ˜ η AD A (values rounded to two decimal places). ˜ K BC = h BC = .12 Pulse Amplitude of IFFLs and Two-Input Circuits at Different Levels of ˜ η AD A Here, a trajectory is considered to contain a pulse if the the trajectory maximum is larger than thepre-induction and post-induction steady states. A trajectory not satisfying this criteria is labeled as"NA".
I1-FFL Type-1 Two-Input Circuit h AD A = K AD A ˜ η AD A h AD A = K AD A ˜ η AD A h AD A = K AD A ˜ η AD A Table S7:
Pulse amplitude of gene C in I1-FFLs and type-1 two-input circuits with different values of ˜ K AD A , h AD A , and ˜ η AD A (values rounded to two decimal places). ˜ K BC = h BC = I4-FFL Type-4 Two-Input Circuit h AD A = K AD A ˜ η AD A h AD A = K AD A ˜ η AD A h AD A = K AD A ˜ η AD A Table S8:
Pulse amplitude of gene C in I4-FFLs and type-4 two-input circuits with different values of ˜ K AD A , h AD A , and ˜ η AD A (values rounded to two decimal places). ˜ K BC = h BC = .13 Comparing the Pulsing Behavior of I1-FFLs and I4-FFLs at Different Levelsof ˜ η BD B Figure S2:
The effect of retroactivity ˜ η BD B on pulse amplitude is more pronounced in an I1-FFL (top row) thanin an I4-FFL (bottom row). Values of the parameters are: ˜ K AB = ˜ K AC = ˜ K BC = ˜ K BD B = h AB = h AC = h BC = h BD B = x C when ˜ η BD B equals0. .14 Response Time of IFFLs with OR Logic at Different Levels of ˜ η BD B Under the assumption of OR logic, the non-dimensionalized ODE model for an I1-FFL without retroac-tivity is given as: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x B d τ = f ˜ B = ¡ − γ B ¢ ‡ ˜ x A ˜ K AB · h AB + ‡ ˜ x A ˜ K AB · h AB + γ B − ˜ x B d ˜ x C d τ = f ˜ C = ¡ − γ C ¢ ‡ ˜ x A ˜ K AC · h AC + ‡ ˜ x A ˜ K AC · h AC + + ‡ ˜ x B ˜ K BC · h BC + γ C − ˜ x C .Under the assumption of OR logic, the non-dimensionalized ODE model for an I4-FFL withoutretroactivity is given as: d ˜ x A d τ = f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + γ A − ˜ x A d ˜ x B d τ = f ˜ B = ¡ − γ B ¢ + ‡ ˜ x A ˜ K AB · h AB + γ B − ˜ x B d ˜ x C d τ = f ˜ C = ¡ − γ C ¢ ‡ ˜ x A ˜ K AC · h AC + ‡ ˜ x A ˜ K AC · h AC + ‡ ˜ x B ˜ K BC · h BC + ‡ ˜ x B ˜ K BC · h BC + γ C − ˜ x C .16 igure S3: In response to an OFF step, the effect of retroactivity ˜ η BD B on response times is more pronounced inan I4-FFL (bottom row) than in an I1-FFL (top row) under the assumption of OR logic. Values of the parametersare: ˜ K AB = ˜ K AC = K BC = ˜ K BD B = h AB = h AC = h BC = h BD B = x C when ˜ η BD B equals 0. I1-FFL I4-FFL h BD B = K BD B ˜ η BD B h BD B = K BD B ˜ η BD B h BD B = K BD B ˜ η BD B Table S9:
Response time of gene C in IFFL models under the assumption of OR logic with different values of˜ K BD B , h BD B , and ˜ η BD B (values rounded to two decimal places). Values of the other parameters are: ˜ K AB = ˜ K AC = h AB = h AC = x C transitions from a high pre-stimulus state to a lowpost-stimulus state in I1-FFLs, I4-FLLs, and their simple regulation counterparts. NA represents cases wherethe post-induction steady state and the mid-point are larger than the pre-induction steady state. .15 Supplemental Information for Figures 4(a) and (b) Response Time argmax t b ( ˜ x B )˜ K BD B ˜ η BD B Table S10:
Response time and time at which b ( ˜ x B ) attains its maximum for the I1-FFL model with h BC = K BD B ˜ η BD B Table S11:
Whether ˜ x B transitions from a value lower (higher) than ˜ K BC to a value higher (lower) than ˜ K BC foran I1-FFL (I4-FFL) model with h BC =
2. Y stands for yes; N stands for no. .16 Proof of the Effects of ˜ η BD B on Response Time and Pulse Amplitude inIFFLs Here we show that for any parameters, increases in retroactivity on node B, ˜ η BD B , in an I1-FFL leadto a decrease in response time and an increase in pulse amplitude.According to Supplemental Information Section 1.6, d ˜ x A d τ , d ˜ x B d τ , and d ˜ x C d τ in an I1-FFL where only˜ η BD B is allowed to vary can be expressed as: d ˜ x A d τ d ˜ x B d τ d ˜ x C d τ = + r ADA ( ˜ x A ) + r BDB ( ˜ x B )
00 0 + r CDC ( ˜ x C ) f ˜ A f ˜ B f ˜ C , (6) f ˜ A = ‡ x I K I A · h I A + ‡ x I K I A · h I A (1 − γ A ) + γ A − ˜ x A f ˜ B = ‡ ˜ x A ˜ K AB · h AB + ‡ ˜ x A ˜ K AB · h AB (1 − γ B ) + γ B − ˜ x B f ˜ C = ‡ ˜ x A ˜ K AC · h AC (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x B ˜ K BC · h BC ¶ (1 − γ C ) + γ C − ˜ x C , (7) r AD A ( ˜ x A ) = ˜ η AD A h AD A (cid:181) ˜ x A ˜ K AD A ¶ h ADA − ˆ + (cid:181) ˜ x A ˜ K AD A ¶ h ADA ! − r BD B ( ˜ x B ) = ˜ η BD B h BD B (cid:181) ˜ x B ˜ K BD B ¶ h BDB − ˆ + (cid:181) ˜ x B ˜ K BD B ¶ h BDB ! − r CD C ( ˜ x C ) = ˜ η CD C h CD C (cid:181) ˜ x C ˜ K CD C ¶ h CDC − ˆ + (cid:181) ˜ x C ˜ K CD C ¶ h CDC ! − , (8)where γ A , γ B , γ C ∈ (0, 1) such that ˜ x A , ˜ x B , ˜ x C ∈ (0, 1).Let ~ ˜ x and ~ ˜ x denote the concentrations of A , B , and C in two I1-FFL models (i.e., ~ ˜ x = [ ˜ x A , ˜ x B , ˜ x C ], ~ ˜ x = [ ˜ x A , ˜ x B , ˜ x C ]), in which all parameters are held identical except that node B is connected todifferent numbers of downstream targets such that retroactivity coefficient ˜ η BD B equals ˜ η BD B and˜ η BD B , respectively. At τ = x I undergoes a stepwise increase and is kept constant afterwards. Theinitial values of ˜ x A , ˜ x B i ( i =
1, 2), and ˜ x C are the corresponding steady values before the increase in x I . Without loss of generality, we assume ˜ η BD B < ˜ η BD B . We now show that ∀ τ >
0, ˜ x C ( τ ) < ˜ x C ( τ ),i.e., the concentration of C in I1-FFL model C in I1-FFL model Lemma 1.
Let d ˜ xd τ = g ( ˜ x )( c − ˜ x ) . If < ˜ x (0) < c, and g ( ˜ x ) is positive and smooth for all ˜ x ∈ (0, ∞ ) , then d ˜ xd τ > for all τ ≥ + .Proof. It is clear that ˜ x has a unique steady state equal to c for ˜ x ∈ (0, ∞ ). Because d ˜ xd τ > x < c and 0 < ˜ x (0) < c , we have d ˜ xd τ > τ ≥ + . Lemma 2.
Suppose that a smooth function h ( τ ) defined on [0, ∞ ) satisfies the following properties: (i)there exists a positive integer k such that d k hd τ k (0 + ) > and d i hd τ i (0 + ) = for all i =
0, 1, 2, ..., k − ; (ii) forany τ ∗ in (0, ∞ ) where h ( τ ∗ ) = we always have dhd τ ( τ ∗ ) > . Then h ( τ ) > for all τ > . roof. Property (i) of h ( τ ) implies that h ( τ ) > δ ). Let (0, T) be the largest intervalwhere h ( τ ) >
0. We claim that T = ∞ . If T < ∞ , then by continuity, h (T) =
0. We immediately arrive ata contradiction as property (ii) implies that h ( τ ) for τ near but less than T cannot be decreasing. Theorem 1. ∀ τ > , ˜ x C ( τ ) < ˜ x C ( τ ) .Proof. We begin by showing that ˜ x A and ˜ x B are monotonically increasing in time. Based on (7), weknow 0 < ˜ x A (0) < ˜ x A ss for nonzero x I . From Lemma 1 it follows that d ˜ x A d τ > τ ≥ + .Let H ˜ A ( ˜ x A ) = ‡ ˜ xA ˜ KAB · hAB + ‡ ˜ xA ˜ KAB · hAB . Assume that there exists τ ∗ ≥ + at which f ˜ B ( τ ∗ ) =
0. Using that f ˜ B ( τ ∗ ) = dH A ( ˜ x A ) d ˜ x A > x A >
0, and d ˜ x A d τ > τ ≥ + , we get d ˜ x B d τ flflflfl τ = τ ∗ = dd τ • + r BD B ( ˜ x B ) f ˜ B ‚flflflfl τ = τ ∗ = dd τ • + r BD B ( ˜ x B ) ‚ · f ˜ B flflflfl τ = τ ∗ + + r BD B ( ˜ x B ) · d f ˜ B d τ flflflfl τ = τ ∗ = + r BD B ( ˜ x B ) · d f ˜ B d τ flflflfl τ = τ ∗ = + r BD B ( ˜ x B ) · • d H A ( ˜ x A ) d τ (1 − γ B ) − d ˜ x B d τ ‚flflflfl τ = τ ∗ = + r BD B ( ˜ x B ) d H A ( ˜ x A ) d ˜ x A d ˜ x A d τ (1 − γ B ) flflflfl τ = τ ∗ > d ˜ x B d τ flflfl τ = + = d ˜ x B d τ flflfl τ = + > d ˜ x B d τ flflfl τ = τ ∗ > d ˜ x B d τ flflfl τ = τ ∗ = τ ∗ >
0, basedon Lemma 2 we know that d ˜ x B d τ > τ > ∀ τ >
0, ˜ x B ( τ ) > ˜ x B ( τ ). Let w B ( τ ) = ˜ x B ( τ ) − ˜ x B ( τ ). Based on (7), we knowthat ˜ x B (0 + ) = ˜ x B (0 + ), i.e., w B (0 + ) =
0. Consider any τ ∗ ≥ + at which w B ( τ ∗ ) =
0, i.e., ˜ x B ( τ ∗ ) = ˜ x B ( τ ∗ ). Because ˜ η BD B < ˜ η BD B , based on (8) we know + r BDB ( ˜ x B ) flflflfl τ = τ ∗ > + r BDB ( ˜ x B ) flflflfl τ = τ ∗ . Hence, d w B d τ flflflfl τ = τ ∗ = dd τ £ ˜ x B − ˜ x B ⁄flflflfl τ = τ ∗ = " + r BD B ( ˜ x B ) f ˜ B − + r BD B ( ˜ x B ) f ˜ B τ = τ ∗ = ˆ + r BD B ( ˜ x B ) − + r BD B ( ˜ x B ) ! f ˜ B flflflflfl τ = τ ∗ =
0, if τ ∗ = + as f ˜ B flfl τ = + = >
0, if τ ∗ > d ˜ x B d τ > τ > =⇒ f ˜ B flfl τ = τ ∗ > τ ∗ = + , we can further show that d w B d τ flflflfl τ = + = d d τ £ ˜ x B − ˜ x B ⁄flflflfl τ = + = ˆ + r BD B ( ˜ x B ) − + r BD B ( ˜ x B ) ! d H A ( ˜ x A ) d ˜ x A d ˜ x A d τ (1 − γ B ) flflflflfl τ = + > w B (0 + ) = dw B d τ flflfl τ = + = d w B d τ flflfl τ = + > dw B d τ flflfl τ = τ ∗ > w B ( τ ∗ ) = τ ∗ >
0, based on Lemma 2 we know that w B ( τ ) >
0, i.e., ˜ x B ( τ ) > ˜ x B ( τ ) for all τ > ∀ τ >
0, ˜ x C ( τ ) < ˜ x C ( τ ). Let w C ( τ ) = ˜ x C ( τ ) − ˜ x C ( τ ). Based on (7), we know˜ x C (0 + ) = ˜ x C (0 + ), i.e., w C (0 + ) =
0. Consider any τ ∗ ≥ + at which w C ( τ ∗ ) =
0, i.e., ˜ x C ( τ ∗ ) = ˜ x C ( τ ∗ ).Let ˜ x C = ˜ x C ( τ ∗ ) = ˜ x C ( τ ∗ ). d w C d τ flflflfl τ = τ ∗ = + r CD C ( ˜ x C ) ¡ f ˜ C − f ˜ C ¢flflflfl τ = τ ∗ .If τ ∗ > + , then ˜ x B ( τ ∗ ) > ˜ x B ( τ ∗ ), which based on (7) indicates that f ˜ C ( τ ∗ ) > f ˜ C ( τ ∗ ). In this case, dw C d τ flflfl τ = τ ∗ > τ ∗ = + . Because ˜ x B (0 + ) = ˜ x B (0 + ) and ˜ x C (0 + ) = ˜ x C (0 + ), we knowthat ∀ n ∈ N , ∂ n ∂ ˜ x ni ‡ d ˜ x C d τ ·flflfl τ = + = ∂ n ∂ ˜ x ni ‡ d ˜ x C d τ ·flflfl τ = + ( i = A , B , B , C , C ). Using the chain rule we can fur-ther show that d w C d τ flflflfl τ = + = dd τ (cid:181) d ˜ x C d τ ¶ − dd τ (cid:181) d ˜ x C d τ ¶flflflfl τ = + = ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ d ˜ x A d τ + ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ d ˜ x B d τ + ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶ d ˜ x C d τ − ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ d ˜ x A d τ − ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ d ˜ x B d τ − ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶ d ˜ x C d τ flflflfl τ = + = ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ d ˜ x A d τ − ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ d ˜ x A d τ flflflfl τ = + = d w C d τ flflflfl τ = + = d d τ (cid:181) d ˜ x C d τ ¶ − d d τ (cid:181) d ˜ x C d τ ¶flflflfl τ = + = dd τ • ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ d ˜ x A d τ ‚ + dd τ • ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ d ˜ x B d τ ‚ + dd τ • ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶ d ˜ x C d τ ‚ − dd τ • ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ d ˜ x A d τ ‚ − dd τ • ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ d ˜ x B d τ ‚ − dd τ • ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶ d ˜ x C d τ ‚flflflfl τ = + = ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ dd τ (cid:181) d ˜ x A d τ ¶ + dd τ • ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶‚ d ˜ x A d τ + ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ dd τ (cid:181) d ˜ x B d τ ¶ + dd τ • ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶‚ d ˜ x B d τ + ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶ dd τ (cid:181) d ˜ x C d τ ¶ + dd τ • ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶‚ d ˜ x C d τ − ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ dd τ (cid:181) d ˜ x A d τ ¶ − dd τ • ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶‚ d ˜ x A d τ − ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ dd τ (cid:181) d ˜ x B d τ ¶ − dd τ • ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶‚ d ˜ x B d τ − ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶ dd τ (cid:181) d ˜ x C d τ ¶ − dd τ • ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶‚ d ˜ x C d τ flflflfl τ = + = ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ d ˜ x A d τ + ∂ ∂ ˜ x A (cid:181) d ˜ x C d τ ¶ (cid:181) d ˜ x A d τ ¶ + ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ d ˜ x B d τ + ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶ d ˜ x C d τ − ∂∂ ˜ x A (cid:181) d ˜ x C d τ ¶ d ˜ x A d τ − ∂ ∂ ˜ x A (cid:181) d ˜ x C d τ ¶ (cid:181) d ˜ x A d τ ¶ − ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ d ˜ x B d τ − ∂∂ ˜ x C (cid:181) d ˜ x C d τ ¶ d ˜ x C d τ flflflfl τ = + = ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ d ˜ x B d τ − ∂∂ ˜ x B (cid:181) d ˜ x C d τ ¶ d ˜ x B d τ flflflfl τ = + .Then because ∂∂ ˜ x B ‡ d ˜ x C d τ ·flflfl τ = + = ∂∂ ˜ x B ‡ d ˜ x C d τ ·flflfl τ = + < d w B d τ flflfl τ = + >
0, we know d w C d τ flflfl τ = + > w C (0 + ) = dw C d τ flflfl τ = + = d w C d τ flflfl τ = + = d w C d τ flflfl τ = + > dw C d τ flflfl τ = τ ∗ > w C ( τ ∗ ) = τ ∗ >
0, based on Lemma 2 we know that w C ( τ ) >
0, i.e., ˜ x C ( τ ) < ˜ x C ( τ ) for all τ > Theorem 2. RT ˜ x C > RT ˜ x C (RT: response time). roof. Based on the previous lemmas and previous theorem, we know that ˜ x C has a unique steadystate ˜ x C ss . Note also that neither ˜ x C ss nor ˜ x C (0) depends on the choice of ˜ η X D X ( X = A , B , C ).Based on Theorem 1, we know that ˜ x C is larger than ˜ x C . This means when ˜ x C reaches the mid-point between ˜ x C (0) and ˜ x C ss (for biological implications, we only consider ˜ x C (0) < ˜ x C ss ), ˜ x C hasreached a value larger than the midpoint. By continuity of ˜ x C , we know that ˜ x C must have reachedthe midpoint earlier than ˜ x C . This in turn implies that the response time of ˜ x C is larger than theresponse time of ˜ x C . 22 .17 Proof of the Effects of Intermodular Retroactivity on Response Time andPulse Amplitude in IFFLs Figure S4:
An I1-FFL connected to n additional circuits, i.e., its context. We now consider the model of an IFFL where node B is connected to, and hence, serves as theinput to other circuits (Figure S4). Using the method shown in Gyorgy and Del Vecchio (2014), wecan derive a model for an IFFL in which intermodular retroactivity is accounted for. d ˜ x A d τ , d ˜ x B d τ , and d ˜ x C d τ in an I1-FFL where only node B is connected to n additional modules (nodes) can be expressedas: d ˜ x A d τ d ˜ x B d τ d ˜ x C d τ = + r BDB ( ˜ x B )1 + r BDB ( ˜ x B ) + Σ ni = S Bi ( ˜ x B )
00 0 1 g ˜ A g ˜ B g ˜ C (9)where S B i ( ˜ x B ) = ˜ η B i h B i (cid:181) ˜ x B ˜ K B i ¶ h Bi − ˆ + (cid:181) ˜ x B ˜ K B i ¶ h Bi ! − (10)and g ˜ A = + r AD A ( ˜ x A ) f ˜ A g ˜ B = + r BD B ( ˜ x B ) f ˜ B g ˜ C = + r BD B ( ˜ x B ) f ˜ C , (11)where r AD A , r BD B , r CD C , f ˜ A , f ˜ B , and f ˜ C are defined the same as in (6).Let ~ ˜ x and ~ ˜ x denote the concentrations of A , B , and C in two I1-FFL models (i.e., ~ ˜ x = [ ˜ x A , ˜ x B , ˜ x C ], ~ ˜ x = [ ˜ x A , ˜ x B , ˜ x C ]), in which all parameters are held identical except that node B is connected to dif-ferent numbers of binding sites in the k -th module of its context (1 ≤ k ≤ n ). ˜ η B k equals ˜ η B k and˜ η B k , respectively. At τ = x I undergoes a stepwise increase and is kept constant afterwards. Theinitial values of ˜ x A , ˜ x B i ( i =
1, 2), and ˜ x C are the corresponding steady values before the increase in x I . Without loss of generality, we assume that for one given k , ˜ η B k < ˜ η B k . We now show that ∀ τ > x C ( τ ) < ˜ x C ( τ ), i.e., the concentration of C in I1-FFL model C inI1-FFL model x A and ˜ x B are monotonically increasing in time. From the proof ofTheorem 1, we already know that d ˜ x A d τ > τ ≥ + . Assume that there exists τ ∗ ≥ + at which d ˜ x B d τ =
0, i.e., g ˜ B =
0. Using that dg ˜ B d τ flflfl τ = τ ∗ > g ˜ B flfl τ = τ ∗ = τ ∗ ≥ + from the proof ofTheorem 1, we get 23 ˜ x B d τ flflflfl τ = τ ∗ = + r BD B ( ˜ x B )1 + r BD B ( ˜ x B ) + Σ ni = S B i ( ˜ x B ) d g ˜ B d τ flflflflfl τ = τ ∗ > d ˜ x B d τ flflfl τ = + = d ˜ x B d τ flflfl τ = + > d ˜ x B d τ flflfl τ = τ ∗ > d ˜ x B d τ flflfl τ = τ ∗ = τ ∗ > d ˜ x B d τ > τ > ∀ τ >
0, ˜ x B ( τ ) > ˜ x B ( τ ). Let w B ( τ ) = ˜ x B ( τ ) − ˜ x B ( τ ). Based on (7) and (9),we know that ˜ x B (0 + ) = ˜ x B (0 + ), i.e., w B (0 + ) =
0. Consider any τ ∗ ≥ + at which w B ( τ ∗ ) =
0, i.e.,˜ x B ( τ ∗ ) = ˜ x B ( τ ∗ ) = ˜ x B ( τ ∗ ). Because ˜ η B k < ˜ η B k , based on (10) we know S B k ( ˜ x B ) flflfl τ = τ ∗ < S B k ( ˜ x B ) flflfl τ = τ ∗ .Hence, d w B d τ flflflfl τ = τ ∗ = ˆ + r BD B ( ˜ x B )1 + r BD B ( ˜ x B ) + Σ ni = i k S B i ( ˜ x B ) + S B k ( ˜ x B ) − + r BD B ( ˜ x B )1 + r BD B ( ˜ x B ) + Σ ni = i k S B i ( ˜ x B ) + S B k ( ˜ x B ) ! g ˜ B flflflflfl τ = τ ∗ =
0, if τ ∗ = + as g ˜ B flfl τ = + = >
0, if τ ∗ > d ˜ x B d τ > τ > =⇒ g ˜ B flfl τ = τ ∗ > τ ∗ = + , we can further show that d w B d τ flflflfl τ = + = ˆ + r BD B ( ˜ x B )1 + r BD B ( ˜ x B ) + Σ ni = i k S B i ( ˜ x B ) + S B k ( ˜ x B ) − + r BD B ( ˜ x B )1 + r BD B ( ˜ x B ) + Σ ni = i k S B i ( ˜ x B ) + S B k ( ˜ x B ) ! · d g ˜ B d τ flflflfl τ = + > w B (0 + ) = dw B d τ flflfl τ = + = d w B d τ flflfl τ = + > dw B d τ flflfl τ = τ ∗ > w B ( τ ∗ ) = τ ∗ >
0, based on Lemma 2 we know that w B ( τ ) >
0, i.e., ˜ x B ( τ ) > ˜ x B ( τ ) for all τ > x C ( τ ) < ˜ x C ( τ ) for all τ > x C is larger than theresponse time of ˜ x C . 24 .18 Proof of the Effects of ˜ η AD A and ˜ η CD C on Response Time in a NegativeAutoregulated Circuit In this section, we show that for any parameters, response time always increases in a negative au-toregulated circuit if retroactivity on node A or C, ˜ η AD A or ˜ η CD C , increases. d ˜ x A d τ and d ˜ x C d τ in a negative autoregulated circuit where only ˜ η AD A may be allowed to vary can beexpressed as: " d ˜ x A d τ d ˜ x C d τ = " + r ADA ( ˜ x A ) + r CDC ( ˜ x C ) f ˜ A f ˜ C , (12) f ˜ A = ‡ x I K I A · h I A + ‡ x I K I A · h I A (1 − γ A ) + γ A − ˜ x A f ˜ C = ‡ ˜ x A ˜ K AC · h AC (cid:181) + ‡ ˜ x A ˜ K AC · h AC ¶ (cid:181) + ‡ ˜ x C ˜ K CC · h CC ¶ (1 − γ C ) + γ C − ˜ x C , (13) r AD A ( ˜ x A ) = ˜ η AD A h AD A (cid:181) ˜ x A ˜ K AD A ¶ h ADA − ˆ + (cid:181) ˜ x A ˜ K AD A ¶ h ADA ! − r CD C ( ˜ x C ) = ˜ η CD C h CD C (cid:181) ˜ x C ˜ K CD C ¶ h CDC − ˆ + (cid:181) ˜ x C ˜ K CD C ¶ h CDC ! − , (14)where γ A , γ C ∈ (0, 1) such that ˜ x A , ˜ x C ∈ (0, 1).Let ~ ˜ x and ~ ˜ x denote the concentrations of A and C in two negative autoregulated circuit mod-els (i.e., ~ ˜ x = [ ˜ x A , ˜ x C ], ~ ˜ x = [ ˜ x A , ˜ x C ]) , in which all parameters are held identical except that nodeA is connected to different numbers of downstream targets such that retroactivity coefficient ˜ η AD A equals ˜ η AD A and ˜ η AD A , respectively. At τ = x I undergoes a stepwise increase and is kept constantafterwards. The initial values of ˜ x A i and ˜ x C i ( i =
1, 2) are the corresponding steady values before theincrease in x I . Without loss of generality, we assume ˜ η AD A < ˜ η AD A . Now we will show that ∀ τ > x C ( τ ) > ˜ x C ( τ ), i.e., the concentration of C in I1-FFL model C in I1-FFL model Theorem 3. ∀ τ > , ˜ x C > ˜ x C and RT ˜ x C < RT ˜ x C (RT: response time).Proof. We begin by showing that ˜ x A and ˜ x C are monotonically increasing in time. Based on (12), weknow ˜ x A (0) < ˜ x A ss for nonzero x I . From Lemma 1 it follows that d ˜ x A d τ > τ ≥ + .Let H ˜ A ( ˜ x A ) = ‡ ˜ xA ˜ KAC · hAC + ‡ ˜ xA ˜ KAC · hAC and H ˜ C ( ˜ x C ) = + ‡ ˜ xC ˜ KCC · hCC . Assume that there exists τ ∗ ≥ + at which f ˜ C ( τ ∗ ) =
0. Because f ˜ C ( τ ∗ ) = dH A ( ˜ x A ) d ˜ x A > x A >
0, and d ˜ x A d τ > τ ≥ + , we get d ˜ x C d τ flflflfl τ = τ ∗ = + r CD C ( ˜ x C ) d f ˜ C d τ flflflfl τ = τ ∗ + dd τ • + r CD C ( ˜ x C ) ‚ f ˜ C flflflfl τ = τ ∗ = + r CD C ( ˜ x C ) • d [ H A ( ˜ x A ) H C ( ˜ x C )] d τ (1 − γ C ) − d ˜ x C d τ ‚flflflfl τ = τ ∗ = + r CD C ( ˜ x C ) • d H A ( ˜ x A ) d ˜ x A d ˜ x A d τ H C ( ˜ x C ) + d H C ( ˜ x C ) d ˜ x C d ˜ x C d τ H A ( ˜ x A ) ‚ (1 − γ C ) flflflfl τ = τ ∗ = + r CD C ( ˜ x C ) d H A ( ˜ x A ) d ˜ x A d ˜ x A d τ H C ( ˜ x C )(1 − γ C ) flflflfl τ = τ ∗ > d ˜ x C d τ flflfl τ = + = d ˜ x C d τ flflfl τ = + > d ˜ x C d τ flflfl τ = τ ∗ > d ˜ x C d τ flflfl τ = τ ∗ = τ ∗ >
0, basedon Lemma 2 we know that d ˜ x C d τ > τ > ∀ τ >
0, ˜ x A > ˜ x A . Let w A ( τ ) = ˜ x A ( τ ) − ˜ x A ( τ ). Based on (13), we know˜ x A (0 + ) = ˜ x A (0 + ), i.e., w A (0 + ) =
0. Consider any τ ∗ ≥ + at which w A ( τ ∗ ) =
0, i.e., ˜ x A ( τ ∗ ) = ˜ x A ( τ ∗ ).Since ˜ η AD A < ˜ η AD A , based on (14) we know + r ADA ( ˜ x A ) flflflfl τ = τ ∗ > + r ADA ( ˜ x A ) flflflfl τ = τ ∗ . Hence, d w A d τ flflflfl τ = τ ∗ = + r AD A ( ˜ x A ) f ˜ A − + r AD A ( ˜ x A ) f ˜ A flflflflfl τ = τ ∗ = " + r AD A ( ˜ x A ) − + r AD A ( ˜ x A ) f ˜ A flflflflfl τ = τ ∗ > w A (0 + ) = dw A d τ flflfl τ = + > dw A d τ flflfl τ = τ ∗ > w A ( τ ∗ ) = τ ∗ >
0, basedon Lemma 2 we know that w A ( τ ) >
0, i.e., ˜ x A ( τ ) > ˜ x A ( τ ) for all τ > ∀ τ >
0, ˜ x C > ˜ x C . Let w C ( τ ) = ˜ x C ( τ ) − ˜ x C ( τ ). Based on (13), we know that˜ x C (0 + ) = ˜ x C (0 + ), i.e., w C (0 + ) =
0. Consider any τ ∗ ≥ + at which w C ( τ ∗ ) =
0, i.e., ˜ x C ( τ ∗ ) = ˜ x C ( τ ∗ ).Because ˜ x A ( τ ) > ˜ x A ( τ ) for all τ >
0, we know H A ( ˜ x A ) > H A ( ˜ x A ) for all τ >
0. Thus, d w C d τ flflflfl τ = τ ∗ = dd τ £ ˜ x C − ˜ x C ⁄flflflfl τ = τ ∗ = + r CD C ( ˜ x C ) f ˜ C − + r CD C ( ˜ x ˜ C ) f ˜ C flflflflfl τ = τ ∗ = + r CD C ( ˜ x C ) ¡ f ˜ C − f ˜ C ¢flflflfl τ = τ ∗ ( =
0, if τ ∗ = + >
0, if τ ∗ > τ ∗ = + , then using dH A ( ˜ x A ) d ˜ x A > x A > dw A d τ flflfl τ = + >
0, and dw C d τ flflfl τ = + =
0, we can further showthat d w C d τ flflflfl τ = + = d d τ £ ˜ x C − ˜ x C ⁄flflflfl τ = + = + r CD C ( ˜ x C ) d H A ( ˜ x A ) d ˜ x A d ˜ x A d τ H C ( ˜ x C )(1 − γ C ) − + r CD C ( ˜ x C ) d H A ( ˜ x A ) d ˜ x A d ˜ x A d τ H C ( ˜ x C )(1 − γ C ) flflflfl τ = + = + r CD C ( ˜ x C ) d H A ( ˜ x A ) d ˜ x A (cid:181) d ˜ x A d τ − d ˜ x A d τ ¶ H C ( ˜ x C )(1 − γ C ) flflflfl τ = + > w C (0 + ) = dw C d τ flflfl τ = + = d w C d τ flflfl τ = + > dw C d τ flflfl τ = τ ∗ > w C ( τ ∗ ) = τ ∗ > w C ( τ ) >
0, i.e., ˜ x C ( τ ) > ˜ x C ( τ ) for all τ > x C is shorter thanthe response time of ˜ x C . 26ow we consider d ˜ x A d τ and d ˜ x C d τ in a negative autoregulated circuit where only ˜ η CD C may be allowedto vary. Let ~ ˜ x and ~ ˜ x denote the concentrations of A and C in two negative autoregulated circuitmodels (i.e., ~ ˜ x = [ ˜ x A , ˜ x C ], ~ ˜ x = [ ˜ x A , ˜ x C ]), in which all parameters are held identical except thatnode C is connected to different numbers of downstream targets such that retroactivity coefficient˜ η CD C equals ˜ η CD C and ˜ η CD C , respectively. At τ = x I undergoes a stepwise increase and is keptconstant afterwards. The initial values of ˜ x A i and ˜ x C i ( i =
1, 2) are the corresponding steady valuesbefore the increase in x I . Without loss of generality, we assume ˜ η CD C < ˜ η CD C . Now we will show that ∀ τ >
0, ˜ x C ( τ ) > ˜ x C ( τ ), i.e., the concentration of C in I1-FFL model C in I1-FFL model Theorem 4. ∀ τ > , ˜ x C > ˜ x C and RT ˜ x C < RT ˜ x C (RT: response time).Proof. Similar to before, we have that ˜ x A and ˜ x C are monotonically increasing in time, i.e., (i) ∀ τ ≥ + , d ˜ x A d τ > ∀ τ > d ˜ x C d τ > ∀ τ >
0, ˜ x C ( τ ) > ˜ x C ( τ ). We define w C ( τ ) and τ ∗ similarly as in the proof ofTheorem 3 such that w C ( τ ∗ ) =
0. Because ˜ η CD C < ˜ η CD C , we have + r CDC ( ˜ x C ) flflflfl τ = τ ∗ > + r CDC ( ˜ x C ) flflflfl τ = τ ∗ .Thus, d w C d τ flflflfl τ = τ ∗ = ˆ + r CD C ( ˜ x C ) − + r CD C ( ˜ x C ) ! f C flflflflfl τ = τ ∗ ( =
0, if τ ∗ = + >
0, if τ ∗ > τ ∗ = + , then using dH A ( ˜ x A ) d ˜ x A > x A > d ˜ x A d τ > τ ≥ + , we can further show that d w C d τ flflflfl τ = + = " + r CD C ( ˜ x C ) − + r CD C ( ˜ x C ) d H A ( ˜ x A ) d ˜ x A d ˜ x A d τ H C ( ˜ x C )(1 − γ C ) flflflflfl τ = + > w C (0 + ) = dw C d τ flflfl τ = + = d w C d τ flflfl τ = + > dw C d τ flflfl τ = τ ∗ > w C ( τ ∗ ) = τ ∗ >
0, based on Lemma 2 we know that w C ( τ ) >
0, i.e., ˜ x C ( τ ) > ˜ x C ( τ ) for all τ >
0. Then similarto the proof of Theorem 2, we conclude that the response time of ˜ x C is shorter than the responsetime of ˜ x C . 27 .19 IFFL Acceleration Persists in the Absence of Parameter Isometry We generated random kinetic parameters via Latin hypercube sampling. To ensure an even distribu-tion over the large space, ˜ K X ( X = AB , AC , BC ) and δ X ( X = A , B , C ) were sampled uniformly on a logscale from the same ranges of values used in Cao et al. (2016) and Shi et al. (2017): ˜ K X ∼ − δ X ∼ −
1. Hill coefficients h X ( X = AB , AC , BC ) were sampled uniformly from a linear intervalstarting at 0.5 and ending at 2, including both positive and negative cooperativity. h BC = 0.5 h BC = 1.0 ̃ η B D B = . h BC = 2.0 h A B ̃ η B D B = . AC ̃ η B D B = . Figure S5:
Percent of I1-FFL trajectories whose relative response time is less than 90% of the model in theabsence of retroactivity calculated based on systematically exploring parameter space as described above, as-suming 0.8 h BC ≤ h BD B ≤ h BC and 0.8 ˜ K BC ≤ ˜ K BD B ≤ K BC . The trajectories are separated evenly by h AB and h AC into 100 voxels, the color of which represents the percent of trajectories whose relative response timeis less than 90% of the model in the absence of retroactivity out of the 100 simulated trajectories falling intothat bin. Here, relative response time is defined as the ratio of the response time of the model to the responsetime of the model without retroactivity. .50.81.11.41.72.0 h BC = 0.5 h BC = 1.0 ̃ η B D B = . h BC = 2.0 h A B ̃ η B D B = . AC ̃ η B D B = . Figure S6:
Median relative response time of I1-FFL trajectories calculated based on systematically exploringparameter space as described above, assuming h BC = h BD B and ˜ K BC = ˜ K BD B . The trajectories are separatedevenly by h AB and h AC into 100 voxels, the color of which represents the median relative response time ofthe 100 simulated trajectories falling into that bin. Here, relative response time is defined as the ratio of theresponse time of the model to the response time of the model without retroactivity. .50.81.11.41.72.0 h BC = 0.5 h BC = 1.0 ̃ η B D B = . h BC = 2.0 h A B ̃ η B D B = . AC ̃ η B D B = . Figure S7:
Percent of I1-FFL trajectories whose relative response time is less than 90% of the model in theabsence of retroactivity calculated based on systematically exploring parameter space as described above, as-suming h BC = h BD B and ˜ K BC = ˜ K BD B . The trajectories are separated evenly by h AB and h AC into 100 voxels,the color of which represents the percent of trajectories whose relative response time is less than 90% of themodel in the absence of retroactivity out of the 100 simulated trajectories falling into that bin. Here, relative re-sponse time is defined as the ratio of the response time of the model to the response time of the model withoutretroactivity. .50.81.11.41.72.0 h BC = 0.5 h BC = 1.0 ̃ η B D B = . h BC = 2.0 h A B ̃ η B D B = . AC ̃ η B D B = . Figure S8:
Median relative response time of I1-FFL trajectories calculated based on systematically exploringparameter space as described above, assuming 0.5 h BC ≤ h BD B ≤ h BC and 0.5 ˜ K BC ≤ ˜ K BD B ≤ K BC . Thetrajectories are separated evenly by h AB and h AC into 100 voxels, the color of which represents the medianrelative response time of the 100 simulated trajectories falling into that bin. Here, relative response time isdefined as the ratio of the response time of the model to the response time of the model without retroactivity. .50.81.11.41.72.0 h BC = 0.5 h BC = 1.0 ̃ η B D B = . h BC = 2.0 h A B ̃ η B D B = . AC ̃ η B D B = . Figure S9:
Percent of I1-FFL trajectories whose relative response time is less than 90% of the model in theabsence of retroactivity calculated based on systematically exploring parameter space as described above, as-suming 0.5 h BC ≤ h BD B ≤ h BC and 0.5 ˜ K BC ≤ ˜ K BD B ≤ K BC . The trajectories are separated evenly by h AB and h AC into 100 voxels, the color of which represents the percent of trajectories whose relative response timeis less than 90% of the model in the absence of retroactivity out of the 100 simulated trajectories falling intothat bin. Here, relative response time is defined as the ratio of the response time of the model to the responsetime of the model without retroactivity. .20 Simulated Synthetic IFFL The I2-FFL model shown in Figure 6 is given as: d x L d t = f L = β L ¡ − γ L ¢ ‡ x I K IL · h IL + ‡ x I K IL · h IL + γ L − δ L x L d x T d t = f T = + η D T h TE x hTE − T K hTETE (cid:181) + ‡ x T K TE · h TE ¶ − β T − γ T + ‡ x L K LT · h LT + γ T − δ T x T d x E d t = f E = β E − γ E (cid:181) + ‡ x L K LE · h LE ¶(cid:181) + ‡ x T K TE · h TE ¶ + γ E − δ E x E ,where L , T , and E represent LmrA, TAL21, and EYFP, respectively. TAL21 is assumed to bind to pUAS-Rep2 (promoter) and D T (decoy sites) with the same affinity and cooperativity, as they share the sameoperator binding sites. η D T is the concentration of the decoy sites of TAL21. The kinetic parametersused in the model are taken from Supplementary Figure 3 in Wang et al. (2019): β L = × MEFL/hr, γ L = × − , δ L = β T = × MEFL/hr, γ T = × − , K LT = × MEFL, h LT = δ T = β E = × MEFL/hr, γ E = × − , K LE = × MEFL, h LE = K T E = × MEFL, h T E = δ E = β X ( X = L , T , E ) are ten times the values givenin Wang et al. (2019) as the rates given in Wang et al. (2019) represent the cell subpopulation withthe lowest production rates. δ X ( X = L , T , E ) are increased for the sake of a faster response time.Biologically, this can be achieved by adding degradation tags to the proteins.33 .21 Two-node Negative Feedback Loops Figure S10:
Two-node negative feedback loops (NFBLs). Left: diagram. Middle: sign-sensitive response accel-eration of a two-node NFBL in the absence of retroactivity, i.e., ˜ η AD A = ˜ η BD B =
0. The response of ˜ x A is acceler-ated in response to an ON step, not in response to an OFF step. “Simple reg.” represents a simple circuit whereA is activated by an external inducer I without additional regulation. The “simple reg.” model achieves thesame steady state as the NFBL model. Right: response times of the two-node NFBL model at different levels of˜ η AD A and ˜ η BD B compared to that of the model with no retroactivity in response to an ON step. The black curve,which we refer to as the “iso-response-time” curve, represents values of ˜ η X D X ( X = A , B ) at which the responsetime is the same as the response time of the model with no retroactivity. Values of parameters used for makingthe middle and the right panels are: ˜ K AB = ˜ K AD A = K B A = ˜ K BD B = h AB = h AD A = h B A = h BD B = Model for the two-node NFBL is: " d ˜ x A d τ d ˜ x B d τ = " + r ADA ( ˜ x A ) + r BDB ( ˜ x B ) f ˜ A f ˜ B , (15) f ˜ A = ¡ − γ A ¢ ‡ x I K I A · h I A + ‡ x I K I A · h I A + ‡ ˜ x B ˜ K BA · h BA + γ A − ˜ x A f ˜ B = ¡ − γ B ¢ ‡ ˜ x A ˜ K AB · h AB + ‡ ˜ x A ˜ K AB · h AB + γ B − ˜ x B , (16) r AD A ( ˜ x A ) = ˜ η AD A h AD A (cid:181) ˜ x A ˜ K AD A ¶ h ADA − ˆ + (cid:181) ˜ x A ˜ K AD A ¶ h ADA ! − r BD B ( ˜ x B ) = ˜ η BD B h BD B (cid:181) ˜ x B ˜ K BD B ¶ h BDB − ˆ + (cid:181) ˜ x B ˜ K BD B ¶ h BDB ! − . (17)34 .22 Significance of Motifs Following the method outlined in Alon (2007), we compared the number of the times an IFFL isobserved in real networks to the number of times an IFFL is expected in a randomized network . Webegan by computing the number of times an IFFL is expected to appear in a randomized ER network.Let G denote a network (graph) consisting of E edges and N nodes. The probability of an edge in agiven direction with the correct interaction type between a pair of nodes is (Alon (2007)): p = E / N ∗ k , (18)where k is the probability that a given edge is positive (activation) or negative (inhibition).According to Alon (2007), the average number of occurrences of an IFFL in the randomized ERnetwork is approximately equal to the number of ways of choosing n nodes out of N times the prob-ability to get g edges with correct interaction types in the correct places: < N G >= N n p g , (19)where both n and g equal 3, since an IFFL contains three nodes and three edges. For convenience ofnotations, we denote the number of occurrences of an IFFL in real networks by ˆ N G .We searched the Regulon database v10.0 (Santos-Zavaleta et al. (2018)) and the TRRUST databasev2 (Han et al. (2018)) for TF-gene interactions in the E. coli (Regulon), mouse (TRRUST), and human(TTRUST) TRNs. The number of genes (nodes), number of edges (interactions), percentage of ac-tivation, percentage of inhibition, and the number of IFFLs are listed in Table S12. Plugging thesevalues into (19), we obtained < N G > .The comparison between real and randomized networks is shown in Table S12. The number ofoccurrences of an IFFL in a real E. coli , mouse, and human TRN is approximately 118.68, 85.61, and161.96 times the number of occurrences of an IFFL in a randomized
E. coli , mouse, and human TRN.Following the same method as above, we compared the number of real and randomized two-nodenegative feedback loops (NFBLs) in different organisms. The number of occurrences of a two-nodeNFBL in a real
E. coli , mouse, and human TRN is approximately 4.82, 18.02, and 18.44 times thenumber of occurrences of a two-node NBFL in a randomized
E. coli , mouse, and human TRN (TableS13).In addition, we found 86 out of 154 inhibitors in
E. coli (Regulon) are negatively auto-regulated,whereas only 5 out of 448 inhibitors in mouse (TTRUST) and 4 out of 470 inhibitors in human (TTRUST)are auto-repressors. The number of occurrences of a negative autoregulatory loop in a real
E. coli ,mouse, and human TRN is approximately 104.88, 6.94, and 4.30 times the number of occurrences ofa negative autoregulatory loop in a randomized
E. coli , mouse, and human TRN (Table S14).
E. coli mouse human N E k + k − N G < N G > ˆ N G < N G > Table S12:
Number of IFFLs in real and randomized
E. coli , mouse, and human TRNs. . coli mouse human N E k + k − N G < N G > ˆ N G < N G > Table S13:
Number of two-node NFBLs in real and randomized
E. coli , mouse, and human TRNs.
E. coli mouse human N E k − N G
86 5 4 < N G > ˆ N G < N G > Table S14:
Number of negative autoregulatory loops in real and randomized
E. coli , mouse, and human TRNs. Supplemental References
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