Origin of Cooperativity in the Activation of Dimeric Transcription Factors

PHYSICAL REVIEW RESEARCH 2, 013151 (2020)

Origin of cooperativity in the activation of dimeric transcription factors

Martin Welch,1 Jens Christian Brasen,2 Christopher T. Workman,3 and Thomas Sams 2,* 1Department of Biochemistry, University of Cambridge, Cambridge CB2 1QW, United Kingdom 2Biomedical Engineering, Technical University of Denmark, Ørsteds Plads 349, 2800 Lyngby, Denmark 3Department of Biotechnology and Biomedicine, Technical University of Denmark, Søltofts Plads 223, 2800 Lyngby, Denmark

(Received 7 March 2019; accepted 12 January 2020; published 12 February 2020)

Cooperative behavior in the binding of ligands to a protein is often viewed as a complex phenomenon where conformational changes induced by the binding of the ﬁrst ligand leads to tighter binding of subsequent ligands. We revisit the ligand-dependent activation of dimeric transcription factors and show that this process may appear cooperative even when it results from independent ligand binding events. This effect is further accentuated through binding of the activated transcription factor to its cognate operator site on the DNA, where we demonstrate that cooperative activation is a stable ﬁxed point. Our analysis nicely accounts for the apparent cooperativity inherent in the biological activity of many dimeric transcription factors.

DOI: 10.1103/PhysRevResearch.2.013151

I. INTRODUCTION factors is dependent upon the binding of low-molecular- weight co-inducers. These small molecules (ligands) are The central dogma of biology describes how genetic in- thought to bind to the transcription factor and alter its con- formation flows from DNA to RNA and thence to protein [1]. formation or oligomeric state, thereby altering the affinity DNA information encoded by genes is therefore expressed via of the transcription factor for the DNA. One may therefore two tightly regulated processes: transcription, where the static express the activity of the transcription factor by the occu- DNA sequence is converted into a transient RNA message, pancy of the operator as a function of ligand concentration. and translation, where the information carried by the RNA Indeed, many transcription factors are now characterized into messenger is decoded to yield functionally active protein. The discrete classes based on the type of ligand bound by the process of transcription is largely controlled by transcription first member of each family to be historically identified. factors [2]. Transcription factors are capable of recognizing We therefore encounter LysR-type or LuxR-type regulators, discrete DNA sequence motifs in regulatory regions called for example. operators located just upstream of the gene(s) of interest. Once Below we will revisit the process of forming active bound to these operator regions, the transcription factor either transcription-factor dimers as a consequence of ligand bind- recruits (if it is acting to stimulate expression of the gene) or ing to the protein and subsequent binding of the activated blocks (if it is acting to repress expression of the gene) access transcription-factor–ligand complex to the DNA. Our analysis of the RNA-synthesizing enzyme, RNA polymerase, to its was inspired by a bacterial cell-cell communication mech- binding site on the DNA. The binding site of RNA polymerase anism called quorum sensing. In quorum sensing, a freely on a gene is known as the promoter and this is usually closely diffusible self-produced signal molecule [frequently, an N- juxtaposed with the operator. This way, binding of a tran- acylated homoserine lactone (AHL)] accumulates in the cul- scription factor to an operator site can alter the transcription ture. Once the AHL concentration exceeds a critical threshold rate of the associated gene. If the gene encodes a protein, the concentration (thought to be determined primarily by binding elevated rate of transcription yields more messenger RNA, of the AHL to a LuxR-type transcription factor), the tran- which in turn yields proportionally more protein following scription factor becomes activated and elicits the transcription translation. of key target genes, which are often involved in bacterial Because of its duplex form, DNA has intrinsic struc- virulence [3,4]. The resulting phenotypic switch is often very tural dyad symmetry. This means that many transcription abrupt and has all the hallmarks of being underpinned by factors function as dimers, with each monomer within the a highly cooperative mechanism [5–8]. However, there is dimer recognizing sequence elements on just one strand of currently no obvious mechanism by which such cooperativity the duplex. Furthermore, the activity of many transcription might be achieved, a problem that we aim to address directly in the present study. Furthermore, our findings can be readily extended to other classes of transcription factor and may well *[email protected] therefore be general. The modeling of quorum sensing systems has been chal- Published by the American Physical Society under the terms of the lenged by the controversy as to whether the LuxR-type Creative Commons Attribution 4.0 International license. Further transcription factors associated with quorum sensing are of distribution of this work must maintain attribution to the author(s) the common form where transcription-factor dimerization and the published article’s title, journal citation, and DOI. drives ligand binding or whether some are of the form where

2643-1564/2020/2(1)/013151(8) 013151-1 Published by the American Physical Society WELCH, BRASEN, WORKMAN, AND SAMS PHYSICAL REVIEW RESEARCH 2, 013151 (2020) ligand binding drives transcription-factor dimerization [9–12]. A. Dimerization drives ligand binding The controversy was partially resolved when Sappington The kinetics describing constitutive production of the tran- et al. [13] demonstrated that, in vivo, LasR folds reversibly scription factor R, the dimerization of the transcription factor, into its dimer form prior to reversibly binding ligands. This and its activation trough ligand S binding are is consistent with the kinetic study reported by Claussen et al. [14]. We will therefore adopt (as a starting point) b λ →1 , →1 , the situation where dimerization of the transcription factor R R (3) precedes ligand binding(s). A formalism for the case where ± k2 λ2 ligand binding precedes dimerization is dealt with in the 2R R2, R2 →, (4) Appendix. ± While looking into the details of the transcription-factor k3 λ + , →3 , activation, we noted that a system driven by continuous R2 S R2S R2S (5) production and turnover of the transcription factor leads to ± k4 λ a rescaled dissociation constant and increased cooperativity. 4 R2S + S R2S2, R2S2 →, (6) In the present work we derive an expression for the effective dissociation constant of the ligand-transcription factor com- as illustrated in Fig. 1. The first step in the process is the plex and establish a proper independence measure for the constitutive production of transcription factors at rate b1. ligand bindings. The formalism clarifies how it is possible to This step encompasses transcription from the DNA segment discriminate between cooperative and noncooperative activa- into messenger RNA as well as translation of the messenger tion and between fast and slow response to changes in ligand RNA into transcription factors. These enzymatic processes concentration, depending on the rate of transcription-factor are carried out by the RNA polymerase and the ribosomes, turnover. respectively. In Fig. 1 the gene segment that codes for the tran- Perhaps the earliest quantitative explanation for ligand scription factor is indicated as a framed R and its associated binding to a protein was proposed by Hill in his analysis promoter PR. The net result of this is the constant production of the oxygen transport properties of haemoglobin. Here the of transcription factors at rate b as indicated in Eq. (3). h/ h + h 1 form s (Kd s ), where s is the ligand concentration, Kd Equations (4)–(6) describe the dimerization of the transcrip- is the dissociation constant, and h is the cooperativity coef- tion factor and activation through ligand bindings into the ficient (also known as the Hill coefficient), provides a good active form R2S2 of the transcription-factor complex. They description of receptor occupancy at ligand concentrations are all reversible reactions between diffusible molecules. On- close to the Kd [15–17]. However, the Hill expression fails to rate constants are superscripted + and off-rate constants are describe the full dynamic range of an activated transcription superscripted −.InFig.1 green arrows indicate the forward factor. In the present work we remedy this by showing that the reactions, i.e., the +. In the first reaction, two transcription concentration of the activated dimer with two ligands bound factors associate with rate constant k+ labeling at both green r = [R S ] can be approximated by the expression 2 4 2 2 arrows leading to the formation of R2, and similarly for the s2 two ligand bindings. The activated transcription-factor com- r = [R S ] = r (1) plex, i.e., R S , is able to bind to its cognate operator region 4 2 2 2 + ζ + 2 4max 2 2 ⎧ Ke 2 eKes s and control the transcription of a gene (X in Fig. 1). In Fig. 1 ⎪ s2 this regulation is indicated by the binding of the activated ⎨⎪ r ,ζ= 0 (cooperative) 2 + 2 4max e transcription factor to the operator P in the regulatory region = Ke s (2) X ⎪ s2 of the gene on the DNA. We will return to this final step ⎩ r ,ζ= 1 (independent), 2 4max e (Ke + s) in Sec. II C. The rightmost processes in Eqs. (3)–(6)arethe turnover of the protein at rates λ which include dilution = i where s [S] is the ligand concentration, Ke denotes by cell division as well as protein degradation by the cell’s the effective dissociation constant (which may differ from recycling system handled by specialized proteins (proteases). the underlying dissociation constant for ligand binding Note that we allow for dissociation of the dimerized transcrip- − to the dimeric transcription factor), and ζe is the independence tion factor R2. Should the dimer be stable, i.e., when k2 is measure, which lies between 0 (cooperative ligand bindings) negligible, the formalism still applies. and 1 (a product of two independent ligand bindings). The If we denote the concentrations by r1 = [R], r2 = [R2], maximal concentration for the active transcription factor r4max r3 = [R2S], r4 = [R2S2], and s = [S], the kinetic equations is reached when the ligand concentration is much larger than corresponding to Eqs. (3)–(6)are the effective dissociation constant. dr 1 = + − − + 2 − λ , b1 2k2 r2 2k2 r1 1r1 (7) II. ANALYSIS dt dr 2 = + 2 + − − + − − − λ , Many transcription factors function as homo- or het- k2 r1 k3 r3 2k3 r2s k2 r2 2r2 (8) erodimers. We will primarily be concerned with the typical dt dr case where homodimerization drives ligand binding. How- 3 = 2k+r s + 2k−r − k+r s − k−r − λ r , (9) ever, for completeness, a brief account of the case where dt 3 2 4 4 4 3 3 3 3 3 ligand binding precedes dimerization is presented in the dr 4 = k+r s − 2k−r − λ r . (10) Appendix. dt 4 3 4 4 4 4

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For independent ligand bindings, r4 deviates by a factor of 4 from its maximal value at the crossing of the asymptotes, while it deviates by a factor of only 2 for cooperative ligand bindings. Its actual deviation at the crossing of the asymptotes will be used to determine the independence measure. The remainder of this section concerns the mathematical details of this procedure. When s is large, r2 and r3 are negligible and r1 assumes its minimal value FIG. 1. Activation of the transcription factor by ligand binding. A schematic diagram shows the production, dimerization, and acti- λ 8b k+ vation of a dimeric transcription factor followed by the binding of r = 1 1 + 1 2 − 1 , (11) 1min 4k+ λ2 the active form of the transcription factor R2S2 to an operator site 2 1 on the DNA controlling the production of substance X. The timeline / = = for forward reactions reads from left to right and following green which can be seen by setting dr1 dt 0 with r2 0. The arrows. Blue arrows signify the time direction for reverse reactions maximal value of r4 is then when appropriate. The subscript number for the rate constants gives b − λ r r = 1 1 1min, (12) the overall time order of reactions corresponding to Eqs. (3)–(6). 4max 2λ Association rate constants are superscripted +, whereas dissociation 4 rates are superscripted −. The monomeric transcription factor is which is just a net dimer source term diluted by growth and denoted by R, while ligands are denoted by S. The different forms degradation λ4. When s is small r1 and r2 will be at their 2 of the transcription factor, i.e., R, R2,R2S, and R2S2, diffuse freely maximal values and r4 will be dominated by the s behavior. as does the ligands. The framed R signifies a piece of DNA that codes In this limit we have for the protein R, while the framed PR is the promoter region on the DNA controlling the transcription of the gene. We now take a quick λ 1 8b λ r = K 1 + 1 2 − , tour through the pathway. (i) The transcription factor R, encoded 1max 2 1 2 1 (13) 2λ2 2 λ K2 by the gene R, is produced at rate b [Eq. (3)]. This involves two 1 1 2 unidirectional processes, first transcription to messenger RNA and r1max r2max = , (14) then translation into the transcription factor. (ii) Two copies of the K2 + − transcription factor bind to each other with rate constant k2 to form k + λ = 2 2 the dimer R2 [Eq. (4)]. (iii) The ligand and the transcription-factor K2 + (15) + k2 dimer associates with rate constant k3 to form R2S [Eq. (5)]. (iv) Another copy of the ligand reacts with the R2S complex to form the and, to leading order in s, we find + activated transcription factor R2S2 with rate constant k [Eq. (6)]. 4 2 2 (v) Finally, the activated transcription factor binds to the operator r1max s r4 = , (16) region controlling the promoter for transcription of the gene X.This K2 K3K4 shifts the production from the background level bx to the induced − + λ − level k . The overall role of the regulatory motif is to control the k3 3 k3 x K3 = + ≈ + , (17) expression of the gene X in response to changing concentrations of k2 k2 the ligand S. At low concentrations of active transcription factors, the − + λ / − k4 4 2 k4 transcription of X proceeds at the background rate bx and shifts to kx K4 = + ≈ + , (18) at high concentrations of active transcription factors. In this paper the k4 k4 = = concentrations (not shown) of constituents are r1 [R], r2 [R2], where K , K , and K are recognized as the dissociation con- = = 2 3 4 r3 [R2S], and r4 [R2S2]. The total concentration of promoter stants for the processes (4)–(6). The asymptotes (12) and (16) = + sitesontheDNAisdefinedaspt [PX ] [R2S2-PX ], while the cross at the effective dissociation constant concentration of activated sites is p = [R S -P ]. All forms of a 2 2 X r the transcription factors may be subject to proteolytic degradation 2 = 4max . Ke K3K4 (19) (not shown). r2max Assuming that dimer forms are equally protected against proteolytic turnover, the effective dissociation constant will The explicit inclusion of the proper statistical weights (factors be larger than the dissociation constant for the second-order 1/2 of 2) allows for simpler interpretation of independence in the binding of the ligand (K3K4 ) . analysis. The effective independence ζe is determined by setting the = The maximal activated concentration r4max, the effective parametrization (1) equal to the steady-state value of r4 at s dissociation constant Ke, and the independence measure ζe Ke. We find the independence term at steady state will now be determined. To determine the ζ = r4max − , effective dissociation constant Ke in the parametrization of e 1 (20) 2r , = the activated transcription factor in Eq. (1), we consider 4 s Ke its asymptotic behavior at small and large values of s for which has the value 1 for independent ligand bindings and 0 the second-order process. On a logarithmic scale, these are for fully cooperative bindings. recognized as straight lines (of slope 2 and 0, respectively) The timescale related to the changes in transcription-factor and the effective dissociation constant sits at their crossing. concentration can be isolated by summing Eqs. (7)–(10).

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Each of the dimeric species contains two transcription-factor C. Cooperativity by binding to the cognate operator (DNA) site constituents and is therefore counted twice. We then arrive at So far, we have investigated the production and activation dr of the transcription factor through ligand binding. Let us now = b − λr, (21) dt 1 consider what happens when the activated transcription factor binds to its cognate operator site on the DNA as indicated in = + + + , r r1 2r2 2r3 2r4 (22) Fig. 1. The concentration of operator sites on the DNA that is occupied by an activated transcription factor is denoted λ r = r λ + 2r λ + 2r λ + 2r λ , (23) 1 1 2 2 3 3 4 4 by pa = [R2S2-PX ] and the total concentration of operator sites is denoted by pt = [PX ] + [R2S2-PX ]. The process is in where r is the total transcription-factor concentration and / λ is the weighted average of the dilution and degradation quasistatic equilibrium (fast on off rates) and therefore the rates. Equation (21) governs the slow changes in the total concentration of occupied DNA sites is transcription-factor concentration that appear when the degra- r4 pa = pt , (29) dation rates differ. Kx + r4 = −/ + where Kx kx kx is the dissociation constant for the pro- B. Independence measure cess. Insertion of r4 from Eq. (1) and rearranging results in We will now try to gain some insight into why underlying 2 s r4max independent ligand bindings do not necessarily lead to non- p = p , (30) a 2 + ζ˜ ˜ + ˜ 2 r + K t cooperative activation. The quasistatic approximations (fast s 2 eKes Ke 4max x / ± λ on off rates ki compared to turnovers i)forEqs.(5) and (6) which is of the same functional form as the activated transcrip- lead to [18] tion factor concentration, but now with appropriately rescaled K K independence measure and dissociation constant r = 3 4 r , (24) 2 K K + 2K s + s2 d 3 4 4 Kx ζ˜e = ζe, (31) 2K4s r + K r = r , (25) 4max x 3 K K + 2K s + s2 d 3 4 4 K K˜ = x K . (32) s2 e r + K e r = r , (26) 4max x 4 K K + 2K s + s2 d 3 4 4 High transcription-factor levels r K will push the = + + 4max x rd r2 r3 r4 (27) activation towards cooperative behavior, i.e., ζ˜e ζe.This criterion is easily satisfied with typical experimentally ob- being satisfied at all times. served dissociation constants and transcription-factor concen- If rd is constant, Eq. (26) is of the same form as Eq. (1) / trations [19]. with effective dissociation constant K = (K K )1 2 and in- √ e 3 4 This way of generating cooperative activation general- dependence measure ζ = K /K . (Note that independent e 4 3 izes to higher-order activation. Furthermore, the activation in ligand bindings have K = K and result in the independence 3 4 downstream first-order processes resulting from the activation measure being 1.) However, if r depends on the ligand d will look increasingly cooperative for each step. In this sense, concentration s, the independence in the underlying reaction cooperative activation constitutes a stable fixed point as a need not be inherited by the overall reaction. We will now look functional form. We express this symbolically as into when this is the case. For simplicity, let us assume that all n n dimer forms of the transcription factor are equally protected s −→ × s , λ = λ = λ = λ const (33) against proteolytic degradation, i.e., 2 3 4 d .In (K + s)n K˜ n + sn the static limit, the total dimer concentration may be written where the arrow signifies convergence for large activating b1 − λ1r1 transcription-factor concentrations or a higher number of first- rd = . (28) 2λd order steps in the sequence of processes. For example, even for fourth-order activation, essentially full cooperativity is − = If the dimers are stable (k2 0), the numerator is a constant easily established with a single activation followed by a first- and so is rd . The effective dissociation constant as well as order process. independence in the underlying reaction is therefore inherited = h/ h + In the literature, the parametrization r4 s (Ke by the overall reaction. sh )r is frequently used as a proxy for the transcription- − > 4max However, if the dimers are unstable (k2 0), the dimer factor–ligand concentration. However, it fails to simultane- concentration in Eq. (28) becomes a function of the lig- ously reproduce the asymptote at s Ke and the behavior and concentration s. In order for Eq. (1) to represent the around s ∼ Ke. Further, it fails to capture the convergence coupled processes, we can remedy this by allowing for a towards cooperative binding as in Eqs. (31) and (33). modified effective dissociation constant and independence as described by Eqs. (19) and (20). Similarly, if the active dimer III. RESULTS AND DISCUSSION has lower proteolytic degradation than the other dimer forms, rd will depend on the ligand concentration and again this leads We have presented two mechanisms that make the sensing to modified effective independence. of a ligand through activation of transcription factors and

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TABLE I. Parameters used in the model where dimerization drives ligand binding in Fig. 2. The two leftmost parameter sets lead to cooperative activation at the promoter as shown in Fig. 2. The low pass set has slow dimer dilution and fast monomer dilution which gives rise to a noise-reducing slow response in transcription-factor activation, i.e., analogous to low-pass ﬁlters. The all pass set has equal dimer and monomer dilution rates which lead to all-pass characteristics, i.e., a fast response. The dilution rates include proteolytic degradation as well as dilution by cell division. The two rightmost parameter sets lead to noncooperative activation at the promoter, one with an unstable dimer and the other with a stable dimer.

Cooperative Noncooperative

Par Low pass All pass Unstable R2 Stable R2 Description Ref. Fig. 2, left Fig. 2, right illustration −1 −1 −1 b1 1000 nM/h 1000 nM h 1000 nM h 1000 nM h constitutive transcription-factor production [20,21] + . −1 −1 . −1 −1 . −1 −1 . −1 −1 → k2 0 05 nM h 0 05 nM h 0 05 nM h 7 2nM h 2R R2 rate constant [21–23] − −1 −1 −1 −1 ← k2 100 h 4000 h 100 h 0h 2R R2 rate [19,21] + −1 −1 −1 −1 −1 −1 −1 −1 + → k3 100 nM h 100 nM h 10 nM h 10 nM h R2 S R2S rate constant [21] − −1 −1 −1 −1 + ← k3 1000 h 1000 h 1000 h 1000 h R2 S R2Srate [21] + −1 −1 −1 −1 −1 −1 −1 −1 + → k4 100 nM h 100 nM h 10 nM h 10 nM h R2S S R2S2 rate constant [21] − −1 −1 −1 −1 + ← k4 1000 h 1000 h 1000 h 1000 h R2S S R2S2 rate [21] −1 −1 −1 −1 λ1 20 h 10 h 0.5h 20 h monomer dilution rate [9,10] −1 −1 −1 −1 λd 0.5h 10 h 0.5h 0.5h dimer dilution rate [11,13,14] r4max 172 nM 19.1 nM 951 nM 847 nM maximal dimer concentration 1/2 (K3K4 ) 10 nM 10 nM 100 nM 100 nM underlying dissociation constant 1/2 (K4/K3 ) 1 1 1 1 underlying independence

Ke 120 nM 125 nM 138 nM 100 nM effective dissociation constant

ζe 0.09 0.15 0.91 1.00 effective independence

subsequent binding to their cognate operator site on the DNA log-linear plots, stressing the behavior around Ke. The log-log appear cooperative. The apparent cooperativity of the acti- plots in Figs. 2(c) and 2(d) provide a representation of the vation of the dimeric transcription factors was found to be full dynamic range of the activation and give a clear view of possible in systems where dimerization drives ligand binding the asymptotic behavior. The parameters used in the calcula- with unstable dimers. The apparent cooperativity generated tions are shown in Table I (indicated as cooperative) and are when activated transcription factors bind to their cognate site all within the referenced physiological ranges. Figures 2(e) on the DNA only requires the maximal transcription-factor and 2(f) show the time response for the two cooperative concentration to be higher than the dissociation constant for parameter sets. This demonstrates that the system is structured the binding to DNA. such that through appropriate evolution of parameters it can Both mechanisms can be seen as a consequence of having effect both slow and fast responses to a sudden increase in the two saturation processes on top of each other. When the ligand concentration. dimer is unstable, the initial balance between the monomer In Table I we also show two examples which display form and the dimer form will be shifted towards the dimer noncooperative behavior in the activation of the transcription form as the ligand concentration increases. This saturation factor. They follow the behavior of the plots with effective process happens in parallel with the underlying shifting be- independence (ζe = 1) in Fig. 2. The left column has unstable tween the R2 form and ligand bound R2S and R2S2 forms of dimers, whereas the right column has stable dimers, thereby the transcription factor. As a result, the effective dissociation covering typical situations. Both parameter sets lead to a fast constant shifts to higher values than the underlying. There- response. We note that it is possible to construct parameter fore, at the effective dissociation constant, the denominator sets with stable dimers and cooperative transcription-factor in Eq. (26) has already shifted to s2 behavior. This leads to activation, though with less typical proteolytic degradation the observed cooperative behavior. With this clarification, we rates (λ2 λ3 = λ4). expect the mechanism to generalize to higher-order activation The cooperativity and the filtering properties of the of transcription factors. The explanation for the cooperative transcription-factor activation depend on the basal produc- behavior as the active transcription factor binds on the DNA is tion of transcription factors. Replacing the promoter driving similar, except here the effective dissociation constant shifts to expression of the transcription factor by a heterologous pro- lower values, but again with the effect that the independence moter with stronger or weaker transcription rate will there- is quenched at ligand concentration approximating to the fore result in altered cooperativity and filtering properties. effective dissociation constant. Similarly, increasing the maximal transcription-factor con- In Fig. 2, two examples with independent underlying centration, e. g., by expressing it from a multicopy plasmid, ligand bindings are shown. As can be seen from the plots will modify the overall cooperativity, the effective disso- in Figs. 2(a)–2(d), both display cooperative behavior in the ciation constant, and the response time of the regulatory activation of the transcription factor. Figures 2(a) and 2(b) are motif.

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(a) ( b)

(e)f)(

FIG. 2. Cooperative activation of the dimeric transcription factor. The activated transcription-factor concentration is shown as a function of ligand concentration for the model where dimerization drives ligand binding with parameter sets from Table I. (a) and (b) Semilogarithmic and (c) and (d) log-log versions are shown. (a), (c), and (e) The slow response and (b), (d), and (f) the fast response are shown. In (a) and (c) and in (b) and (d) we observe that transcription-factor activation appears almost fully cooperative even though the underlying ligand bindings are n independent. (c) and (d) Time response following introduction of the ligand at concentrations s = 2 Ke, n =−3, −2,...,5.

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We have given a couple of typical examples of regulatory with combinatorial weights explicitly included. We employ systems which display cooperative behavior without requiring the convention that reactions are numbered by the number of cooperativity at the elementary level. We expect that simple constituents in the resulting product. mechanisms similar to those presented here may apply to Again, in the static limit of the activation, the form (1) other regulatory systems as well. provides an excellent description. To determine Ke we need the asymptotic behavior at small s and at large s in the static limit of the kinetic equations. The derivation is analogous to IV. CONCLUSION the derivation of Eqs. (11)–(19). With dissociation constants deﬁned as The primary ﬁnding from our analysis is that the coopera- − − tive activation of transcription factors can be established even k + λ2 k K = 2 ≈ 2 , (A7) with independent underlying ligand bindings. The binding 2 + + k2 k2 of the activated transcription-factor complex to its cognate − + λ − k4 4 k4 operator site on the DNA pushes the regulation even further in K4 = + ≈ + , (A8) the direction of cooperative behavior. Our analysis accounts k4 k4 for the inherent cooperativity in the biological activity of the asymptotic behavior at small s is given by many dimeric transcription factors. r2 b2 A number of advantages are associated with the cooper- r = 1max s2 = 1 s2, (A9) 4 2 λ2 2 ative activation of transcription factors and most are related K2 K4 1K2 K4 to improvements of basic sensory function. Cooperative ac- b r = 1 . tivation can provide an expansion of the dynamic range of 1max λ (A10) gene expression and it can improve stability when occurring 1 in feedback systems that control rapid switching between The maximal RS and activated transcription-factor levels are different responses. It is therefore important that we have reached at large s: modeling approaches that can account for cooperativity in λ 8b λ sensory systems where underlying cooperative mechanisms r = 2 K 1 + 1 4 − 1 , (A11) 2max λ 4 λ2 may not be present, thus explaining cooperativity without the 4 4 K4 2 need of elaborate evolutionary constraints. 2 r2max r4max = . (A12) K4

APPENDIX: LIGAND BINDING DRIVES DIMERIZATION The effective dissociation constant sits at the crossing of the asymptotes, For completeness, we include the simplest version of a K2 r2max regulatory motif where ligand binding drives dimerization Ke = K4r4max = K2 , (A13) r1max r1max and is typically different from K . b1 λ1 2 → R, R → (A1) In the limit of low- or high-level production of transcription ± factors we find k2 λ2 ⎧ R + S RS, RS → (A2) λ2 λ ⎨ 1 2, 8b1 4 λ2 K2 K λ2 1 ± 2 = 2 4 2 k4 λ K (A14) 4 e ⎩ K λ2 λ 2(RS) R S , R S →. (A3) 4 1 2, 8b1 4 . 2 2 2 2 λ K2 λ2 1 2 4b1 K4 2 In the case where the production of transcription factors is low The corresponding kinetic equations are and the monomer forms have similar degradation rates, this does not lead to a modification of the dissociation constant. dr However, when there is large production of transcription fac- 1 = b + k−r − k+r s − λ r , (A4) dt 1 2 2 2 1 1 1 tors, the rescaling of the dissociation constant can be signifi- cant. In the case where ligand binding drives dimerization, we dr 2 = + + − − + 2 − − − λ , k2 r1s 2k4 r4 2k4 r2 k2 r2 2r2 (A5) did not find cooperativity in the activation with physiological dt values of the parameters that we tested. However, the tran- dr 4 = k+r2 − k−r − λ r , (A6) scriptional activation at the promoter may still be cooperative dt 4 2 4 4 4 4 as a consequence of the mechanism described in Sec. II C.

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