Kinetics of Protein-DNA Interaction: Facilitated Target Location in Sequence-Dependent Potential

Transcription

1 Biophyical Journal Volume 87 December Kinetic of Protein-DNA Interaction: Facilitated Target Location in Sequence-Dependent Potential Michael Slutky* and Leonid A. Mirny* y *Department of Phyic and y Harvard-MIT Diviion of Health Science and Technology, Maachuett Intitute of Technology, Cambridge, Maachuett ABSTRACT Recognition and binding of pecific ite on DNA by protein i central for many cellular function uch a trancription, replication, and recombination. In the proce of recognition, a protein rapidly earche for it pecific ite on a long DNA molecule and then trongly bind thi ite. Here we aim to find a mechanim that can provide both a fat earch (1 10 ) and high tability of the pecific protein-dna complex (K d ¼ 10 ÿ15 10 ÿ8 M). Earlier tudie have uggeted that rapid earch involve liding of the protein along the DNA. Here we conider liding a a one-dimenional diffuion in a equencedependent rough energy landcape. We demontrate that, depite the landcape roughne, rapid earch can be achieved if one-dimenional liding i accompanied by three-dimenional diffuion. We etimate the range of the pecific and nonpecific DNA-binding energy required for rapid earch and ugget experiment that can tet our mechanim. We how that optimal earch require a protein to pend half of it time liding along the DNA and the other half diffuing in three dimenion. We alo etablih that, paradoxically, realitic energy function cannot provide both rapid earch and trong binding of a rigid protein. To reconcile thee two fundamental requirement we propoe a earch-and-fold mechanim that involve the coupling of protein binding and partial protein folding. The propoed mechanim ha everal important biological implication for earch in the preence of other protein and nucleoome, imultaneou earch by everal protein, etc. The propoed mechanim alo provide a new framework for interpretation of experimental and tructural data on protein-dna interaction. INTRODUCTION The complex trancription machinery of cell i primarily regulated by a et of protein, trancription factor (TF), that bind DNA at pecific ite. Every TF can have from one to everal dozen of uch pecific ite on the DNA. Upon binding to the ite, TF form a table protein-dna complex that can either activate or repre trancription of nearby gene, depending on the actual control mechanim. Fat and reliable regulation of gene expreion require 1), fat (;1 10 ) earch and recognition of the pecific ite (referred to a the target or cognate ite below) out of poible ite on the DNA; and 2), tability of the protein-dna complex (K d ¼ 10 ÿ15 10 ÿ8 M). Depite it apparent implicity, uch a mechanim i not undertood in depth, either qualitatively or quantitatively. Here we focu on a impler cae of bacterial TF recognizing their cognate (target) ite on the naked DNA. Needle to ay, eukaryotic protein-dna recognition i ignificantly complicated by chromatin packing of the DNA and the multiubunit tructure of the TF. Interetingly, imilar problem of pecific binding and binding rate arie in the context of oligonucleotide-dna binding (Lomakin and Frank-Kamenetkii, 1998). hundred of mutated protein (Takeda et al., 1989; Grillo et al., 1999), calorimetry meaurement (Spolar and Record, 1994), and novel ingle-molecule experiment (Shimamoto, 1999). Thee experimental data contributed mot ignificantly to our preent undertanding of protein-dna interaction ince the early work of von Hippel and co-worker. In a erie of pioneering article (Berg et al., 1981; Winter et al., 1989; von Hippel and Berg, 1989; Berg and von Hippel, 1987), they created a conceptual bai for decribing both the kinetic and thermodynamic of protein-dna interaction, which ha ince become a tarting point for practically every ubequent theoretical work on the ubject. We tart by reviewing the hitory of the problem and decribing the paradox of the fater-than-diffuion aociation rate. Next, we preent the claical model of protein- DNA liding and explain how thi model can reolve the paradox. We outline the problem that the liding mechanim face if the energetic of protein-dna interaction are taken into account. Next we introduce our novel quantitative formalim and undertake an in-depth exploration of poible mechanim of protein-dna interaction. Vat amount of experimental data available thee day provide the tructure of protein-dna complexe at atomic reolution in crytal and in olution (Lucombe et al., 2000; Fater-than-diffuion earch Bell and Lewi, 2001, 2000; Lewi et al., 1996; Schumacher The problem of how a protein find it target ite on DNA ha et al., 1994), binding contant for dozen of native and a long hitory. In 1970, Rigg et al. (1970a,b) meaured the aociation rate of LacI repreor and it operator on DNA a ;10 10 M ÿ1 ÿ1 Submitted July 29, 2004, and accepted for publication September 15, Thi atonihingly high rate (a compared to Addre reprint requet to Leonid A. Mirny, Tel.: ; other biological binding rate) wa hown to be much higher leonid@mit.edu. than the maximal rate achievable by three-dimenional Ó 2004 by the Biophyical Society /04/12/4021/15 $2.00 doi: /biophyj

2 4022 Slutky and Mirny diffuion. In fact, if a protein bind it ite by threedimenional diffuion, it ha to hit the right ite on the DNA within b ¼ 0.34 nm. A hift by 0.34 nm would reult in binding a ite that i different from the native ite by 1 bp. Such a ite can be very different; e.g., GCGCAATT veru CGCAATTC. Uing the Debye-Smoluchowki equation for the maximal rate of a bimolecular reaction (ee e.g., Richter and Eigen, 1974; Flyvbjerg et al., 2002; Bruinma, 2002), with a protein diffuion coefficient of D 3d ;10 ÿ7 cm 2 ÿ1 (Elowitz et al., 1999) we get k DS ¼ 4pD 3D b ; 10 8 M ÿ1 ÿ1 : (1) Thi value for the aociation rate, relevant for in vitro meaurement, correpond to target location in vivo on a timecale of a few econd, when each cell contain up to everal ten of TF molecule. To reolve the dicrepancy between the experimentally meaured rate of M ÿ1 ÿ1 and the maximal rate of 10 8 M ÿ1 ÿ1 allowed by diffuion, Richter and Eigen (1974), and later Berg et al. (1981) and von Hippel and Berg (1989), uggeted that the dimenionality of the problem change during the earch proce. They concluded that, while earching for it target ite, the protein periodically can the DNA by liding along it. Sliding along the DNA If a protein perform both three-dimenional and onedimenional diffuion, then the total earch proce can be conidered a a three-dimenional earch followed by binding DNA and a round of one-dimenional diffuion. Upon diociation from the DNA, the protein continue three-dimenional diffuion until it bind DNA in a different place, and o on. Some experimental evidence upport thi earch mechanim. Thee include affinity of the DNAbinding protein for any fragment of DNA (nonpecific binding), ingle molecule experiment where one-dimenional diffuion ha been oberved and viualized, and numerou other experiment where the rate of pecific binding to the target ite ha been ignificantly increaed by lengthening nonpecific DNA urrounding the ite (Kim et al., 1987). What are the benefit and the mechanim of one-dimenional diffuion and what limit the earch rate? Here we addre thi quetion and conider poible earch mechanim that involve both one-dimenional and three-dimenional diffuion, where one-dimenional diffuion along the DNA proceed along the rough energy landcape. Quantitative analyi of the earch proce brought u to the following four main reult: 1. When the roughne of the binding energy landcape i *2 k B T, the diffuion along the DNA become extremely low, with the protein unable to diffue more than a few baepair. The total earch proce i prohibitively low. 2. If the earch proceed by a combination of onedimenional and three-dimenional diffuion, nonpecific binding to the DNA play a very important role in controlling the balance between thee two procee. The optimal energy of nonpecific binding can provide the maximal earch rate. Although fater than either threedimenional or one-dimenional earch alone, optimal combination of three-dimenional and one-dimenional diffuion cannot expedite the earch if the roughne of the landcape i *2 k B T. 3. Experimentally oberved and biologically relevant rate of earch can be reached only when one-dimenional liding proceed through a fairly mooth landcape with a roughne of the order of k B T. 4. Paradoxically, the tability of the protein-dna complex at the target ite require a roughne of the binding energy landcape coniderably larger than k B T. Rapid earch, however, by one-dimenional/three-dimenional diffuion i impoible at uch a roughne. Finally, we formulate thi earch-peed/tability paradox and ugget a earch-and-fold mechanim that can reolve it. The paradox can be reolved if the DNA-binding protein ha two ditinct (conformational) tate in which it exhibit two mode of binding. In the firt, which i the mode that ha weaker binding and a moother landcape, it earche for it ite. In the econd (recognition) mode, which ha larger roughne of the binding landcape, the protein tightly bind DNA ite. Correlation between the energy landcape in the two mode and the energy difference and the barrier between the two protein conformation control the frequency of tranition between the two mode and provide effective preelection of low-energy ite. We ugget that thee mode correpond to two ditinct conformational tate of the protein-dna complex (a relatively open complex in the earch mode, and a tighter complex in the recognition mode). Tranition between the two tate can include partial folding of the protein, water extruion, change in the DNA conformation, etc. Focuing on the conformation of the protein, and without lo of generality, we conider a partially unfolded (diordered) conformation and the folded conformation bound to the cognate ite a the two conformation required by our model. In fact, a protein in the partially unfolded conformation may have fewer and/or weaker interaction with DNA allowing rapid liding. Folded conformation, in turn, provide tronger and more pecific interaction required for tight binding. We alo quantify the requirement of thi two-mode mechanim to provide both rapid earch and tability. Structure of known DNA-binding protein are known to be flexible and have been reported to exhibit two or more ditinct binding mode. Thi two-tate mechanim alo agree well with the reult of calorimetric experiment. The propoed earch-and-fold mechanim i not limited to the protein-dna interaction; it alo provide a general

3 Kinetic of Protein-DNA Interaction 4023 framework for protein-ligand binding and demontrate the advantage of induced folding, a common theme in molecular recognition. THE MODEL Search time In our model, the earch proce conit of N round of one-dimenional earch (each take time of t 1d,i, i ¼ 1...N) eparated by round of threedimenional diffuion (t 3d, i ). The total earch time t i the um of the time of individual earch round, N t ¼ + ðt 1d;i 1 t 3d;i Þ: (2) i¼1 The total number N of uch round occurring before the target ite i eventually found i very large, o it i natural to introduce probability ditribution for the eentially random entitie in the problem. The firt obviou implification that can be made without any lo of rigor i to replace t 3d,i by it average t 3d. Each round of one-dimenional diffuion can a region of n ite (where n i drawn from ome ditribution p(n)). The time, t 1d (n), that it take to can n ite can be obtained from the exact form of the one-dimenional diffuion law (ee Appendix A). If, on average, n ite are canned in each round, then the average number of uch round required to find the ite of length M on DNA i N ¼ M=n: Uing average value, we get a total earch time of t ðn; MÞ ¼ M n ½t 1dðnÞ 1 t 3d Š: (3) From Eq. 3 it i clear that, in general, t ðn; MÞi large for both very mall and very large value of n: In fact, if n i mall, o few ite are canned in each round of the one-dimenional earch that a large number of uch round (alternating with round of three-dimenional diffuion) are required to find the ite. On the other hand, if n i large, lot of time i pent canning a ingle tretch of DNA, making the earch very redundant and inefficient. An optimal value, n opt ; hould exit, which provide little redundancy of onedimenional diffuion and a ufficiently mall number of uch round. For a given diffuion law t 1d (n), function t ðn; MÞ can be minimized producing n opt ; the optimal length of DNA to be canned between the aociation and the diociation event. (Naturally, we aume here that t 1d ðnþ grow with n at leat a Oðn 11a Þ; with a. 0.) Protein-DNA energetic While diffuing along DNA, a TF experience the binding potential Uð~Þ of every ite ~it encounter. The energy of protein-dna interaction i uually divided into two part pecific and nonpecific (Berg and von Hippel, 1987; Gerland et al., 2002), U i ¼ Uð~¼ i ;... i1lÿ1 Þ 1 E n ; (4) where ~ decribe a binding DNA equence of length l. A it name ugget, the nonpecific binding energy E n arie from interaction that do not depend on the DNA equence that the TF i bound to, e.g., interaction with the phophate backbone. The pecific part of the interaction energy exhibit a very trong dependence on the actual nucleotide equence. Here and below we ue the term energy to refer to the change in the free energy related to binding DG b. Thi free energy include the entropic lo of tranlational and rotational degree of freedom of the protein and amino acid ide chain, the entropic cot of water and ion extruion from the DNA interface, the hydrophobic effect, etc. The energy of pecific protein-dna interaction can be approximated by a weight matrix (alo known a PSSM, or profile) where each nucleotide contribute independently to the binding energy (Berg and von Hippel, 1987), l Uð~¼ i ;... i1lÿ1 Þ¼+ eðj; j Þ; (5) j¼1 where j i a baepair in poition j of the ite and e(j, x) i the contribution of baepair x in poition j. Mot of the known weight matrice of TF e(j, j ) give rie to uncorrelated energie of overlapping neighboring ite, obtained by one baepair hift (Gerland et al., 2002). Fig. 1 preent ditribution of the equence pecific binding energy f(u) obtained for different bacterial trancription factor and all poible ite in the correponding genome. The weight matrice for thee trancription factor ha been derived uing a et of known binding ite and tandard approximation (Berg and von Hippel, 1987; Stormo and Field, 1998). Notice that for a ufficiently long ite the ditribution of the binding energy of random ite (or genomic DNA) can be cloely approximated (ee Fig. 1) by a Gauian ditribution with a certain mean hui and variance 2, f ðu i Þ¼ ffiffiffiffiffiffiffiffiffiffi 1 p exp ÿ ðu i ÿhuiþ 2 : (6) 2p We alo aume independence of the energy of neighboring (although overlapping) ite. Binding energie calculated for bacterial TF upport thi aumption. Other phyical factor uch a local DNA flexibility (Erie et al., 1994) can create a correlated energy landcape, providing a different mode of diffuion, a we have decribed in Slutky et al. (2004). Diffuion in a equence-dependent energy landcape The whole DNA molecule can thu be mapped onto a one-dimenional array of ite, f~ i g each correponding to a certain binding equence compriing bae from the i th to the (i 1 l 1) th, l being the length of the motif (ee Fig. 2). At each ite, there i a probability p i of hopping to ite i 1 1 and a probability q i of hopping to ite i 1. Thee probabilitie depend on the pecific binding energie, U i and U i61, at the i th ite and at the adjacent ite, repectively, and are proportional to the correponding tranition rate, v i,i11 and v i,i 1. For the latter, it i mot natural to aume the regular activated tranport form v i;i61 ¼ n 3 eÿbðu i61ÿu i Þ if U i61. U i ; (7) 1:0 otherwie where n i the effective attempt frequency, b [ (k B T) ÿ1 ; k B i the Boltzmann contant; and T i the ambient temperature. Having defined that, we have a one-dimenional random walk with poition-dependent hopping probabilitie. A ha been hown in numerou article throughout the lat two decade, the propertie of one-dimenional random walk can vary dramatically depending on the actual choice of probabilitie, fp i g (for review, ee Bouchaud and George, 1990). Here we employ the mean firt-paage time formalim (Murthy and Kehr, 1989) to derive the diffuion law t 1d ðnþ for protein liding along the DNA given the equence-dependent binding energy (Eq. 7). RESULTS Uing the model decribed above, we tudied the following problem: 1. How fat i the one-dimenional earch on DNA a a function of the roughne,, of the binding energy landcape?

4 4024 Slutky and Mirny exhibit an exponential dependence on the roughne of the binding energy landcape, dropping rapidly a become greater than a few k B T (Slutky et al., 2004). Hence, rapid diffuion of a protein along the DNA i poible only if the roughne of the binding energy landcape i mall compared to k B T (b, 1.5). Thi requirement impoe trong contraint on the allowed energy of pecific binding interaction. FIGURE 1 Spectrum of binding energy for three different trancription factor and the Gauian approximation (olid line). 2. How ignificant i the role of nonpecific binding energy, E n, in determining the earch time? 3. How fat i the earch for the native ite under condition that provide tability to the protein-dna complex at the target ite? Diffuion along the DNA We tate here the main reult without a derivation (which can be found in Appendix A). For a given et of probabilitie fp i g, the mean firt-paage time (MFPT) from i ¼ 0toi¼L (in term of number of tep) i (Murthy and Kehr, 1989) Lÿ1 t 0;L ¼ L 1 + k¼0 Lÿ2 a k 1 + Lÿ1 + k¼0 i¼k11 ð1 1 a k Þ Yi j¼k11 a j ; (8) where a i [ q i /p i. The relation in Eq. 8 give the MFPT for one given realization of probabilitie. Auming that the pecific binding energie fu i g have a normal ditribution with variance 2 (ee above), we plug the probabilitie in Eq. 7 into Eq. 8 and after a omewhat lengthy but traightforward calculation, we obtain an expreion for the MFPT averaged over genomic equence for L 1, ht FP ðlþi t 0 L 2 e 7b2 2 =4 ð1 1 b 2 2 =2Þ ÿ1=2 ; (9) where t 0 i the reciprocal of the effective attempt frequency for hopping to a neighboring ite. The main reult i that the one-dimenional earch by hopping to neighboring ite proceed by normal diffuion with t ; L 2 /2D 1d, where the diffuion coefficient D 1d ðþ b2 2 1=2 e ÿ7b2 2 =4 (10) 2t 0 2 Optimal time of three-dimenional/ one-dimenional earch When one-dimenional canning i combined with threedimenional diffuion, what i the optimal time a protein ha to pend in each of the two regime? To anwer thi quetion we compute the optimal number of ite the protein ha to can by one-dimenional diffuion to get the fatet overall earch. Reult of thi ection are rather general and are not limited to the particular cenario of low one-dimenional diffuion on a rough landcape dicued above. Each time the protein bind DNA it perform a round of one-dimenional diffuion. If the p round lat t 1d, then, on average, the protein can n ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16D 1d t 1d =p bp (Hughe, 1995). By plugging thi relation into Eq. 3 for earch time t, and minimizing t with repect to n; we get the optimal total earch time and the optimal number of ite to be canned in each round, t opt ¼ t ðn opt Þ¼ M rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pt 3d 16 n opt ¼ 2 D 1d p D 1dt 3d ; (11) which bring u to the following concluion. Firt, and mot importantly, we obtain that, in the optimal regime of earch, t 1d ðn opt Þ¼t 3d ; (12) i.e., the protein pend equal amount of time diffuing along nonpecific DNA and diffuing in the olution. Thi triking reult i very general, and i true irrepective of the value of diffuion coefficient D 1d or D 3d, or ize of the genome M.In p fact it follow directly from the diffuion law n ; ffiffiffiffiffiffi t 1d: More importantly thi central reult can be verified experimentally by either ingle-molecule technique or by traditional method. Alo note that the optimal region of the DNA canned in a ingle round of one-dimenional diffuion n opt doe not depend on M i.e., i the ame irrepective of the ize of the genome to be earched for a pecific ite. Second, the optimal one-dimenional/three-dimenional combination reached at t 1d ¼ t 3d lead to a ignificant peedup of the earch proce. In fact, an optimal onedimenional/three-dimenional earch i n opt time fater than a earch by three-dimenional diffuion alone, and M=n opt time fater than a earch by one-dimenional diffuion alone. For example, if the protein operate in the

5 Kinetic of Protein-DNA Interaction 4025 FIGURE 2 The model potential. optimal one-dimenional/three-dimenional regime and can n opt ¼ 100 bp during each round of DNA binding, then the experimentally meaured rate of binding to the pecific ite can be 100 time greater than the rate achievable by three-dimenional diffuion alone. Third, we can etimate n opt ; the maximal number of ite a protein can can in each round of one-dimenional earch. If we et D 1d to it maximum, i.e., D 1d ; D 3d and t 3d ;l 2 d =D 3d; with l m ; 0.1 mm, we get n max opt ; 500 bp: (13) For a maller one-dimenional diffuion coefficient, e.g., D 1d ; D 3d /100, we get n max opt ;50 bp: Again, ingle molecule experiment can provide etimate of thee quantitie for different condition of diffuion. Finally, we obtain etimate of the hortet poible total earch time. If M 10 6 bp and one-dimenional diffuion i at it fatet rate, i.e., D 1d ; D 3d ¼ 10 ÿ7 cm 2 /, then uing Eq. 11 we get ; M pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pt 3d t 0 ; 5; (14) 2 t opt where we etimate t 0 ;a 2 0 =D 3d;10 ÿ8 : One can alo etimate the earch time uing in vitro experimentally meaured binding rate in water, kon water M ÿ1 ÿ1 (Rigg et al., 1970a,b). The diffuion coefficient of a protein in the cytoplam i time lower than that in water, leading to the etimated binding rate of kon cytoplam 10 8 ÿ 10 9 M ÿ1 ÿ1 (ee Appendix D). From thi we obtain the time it take for one protein to bind one ite in a cell of 1 mm 3 volume (i.e., [TF] 10 ÿ9 M) a t exp ¼ðk cytoplam on ½TFŠÞ ÿ1 ; 1ÿ10 : (15) One can ee perfect agreement between our theoretical etimate and experimentally meaured binding rate. A we mentioned above, there are uually everal TF molecule earching in parallel for the target ite. Naturally, in thi cae, the earch i ped up proportionally to the number of molecule. Diffuion of PurR on the Echerichia coli genome To check the applicability of the above conideration, we imulated one-dimenional diffuion of PurR trancription factor on the E. coli chromoome. The pecific energy profile wa built uing a weight matrix derived from 35 PurR binding ite following a tandard procedure decribed elewhere (Berg and von Hippel, 1987; Stormo and Field, 1998). The reulting energy profile i random and uncorrelated and ha a tandard deviation 6.5 k B T. Thi profile wa ued a an input for calculating mean firt paage time at different temperature. (Since the magnitude of the interaction i fixed, in thee calculation we vary temperature rather than binding trength.) The reult of thee calculation i preented in Fig. 3. It i clear that when the roughne of the landcape become ignificant at

6 4026 Slutky and Mirny. 2 k B T, the diffuion proceed extremely lowly. Only ; bp can be canned by a TF when ¼ 2 k B T.A natural requirement for ufficiently fat diffuion i, a before, ; k B T. Nonpecific binding Wherea the diffuion of the TF molecule along DNA i controlled by the pecific binding energy, the diociation of the TF from the DNA depend on the total binding energy, i.e., on the nonpecific binding a well a on the pecific one. Moreover, ince the diociation event are much le frequent than the hopping between neighboring baepair (roughly by a factor of t 3d =hti), the nonpecific energy E n make a enibly larger contribution to the total binding energy. For a TF at ret bound to ome DNA ite i, the diociation rate, r i, would be given by the Arrheniu-type relation, r i ¼ 1 t 0 e ÿbðenÿu iþ : (16) Given the pecific (U i ) and the nonpecific (E n ) energy, one can calculate the average time, t 1d, a protein pend before diociating from the DNA (ee Appendix B). We obtain " E n ¼ k B T ln t 1d ÿ 1 # 2 ; (17) t 0 2 k B T and in the optimal regime where t 1d ¼ t 3d ; " E opt n ¼ k B T ln t 3d ÿ 1 # 2 : (18) t 0 2 k B T The parameter pace Since for a given value of, the nonpecific binding control the diociation rate, the earch time will deviate from the optimum if E n move from thi predetermined value. In Fig. 4 a we plot the earch time a a function of the nonpecific binding energy for different value of. We now define the tolerance factor, z, a the ratio between the acceptable value of the earch time, t, and the optimal earch time, t opt : Experimental data ugget z # 5, but for the moment we allow for much larger value of z ; (thi can be done when, for intance, there are many protein molecule earching in parallel). A we can ee from Fig. 4 a, for each value of, there i a range of poible value of E n uch that the reulting earch time i within the region of tolerance (ee Appendix B). Note the dramatic increae in the earch time a E n deviate from it optimal value. Specifying z, we can define our parameter pace, i.e., the value of pecific and nonpecific energy producing a total earch time within the region of tolerance. In Fig. 4 b, we conider three value of z. The mot relaxed requirement z ¼ 100 provide a earch time of t # 500. If 100 protein are earching for a ingle ite, then the firt one will find it after FIGURE 3 The mean firt paage time veru traveling ditance for purr trancription factor on the binding landcape of different roughne (or at different temperature). The horizontal line indicate the optimal regime, t 1d ; t 3d : ;5 leading, however, to a fairly low binding rate of k on 1/ M ÿ1 ¼ M ÿ1 ÿ1 (compared to experimentally meaured M ÿ1 ÿ1 in water). Importantly, to comply with even thi mot relaxed earch time requirement, the characteritic trength of pecific interaction mut be &2.3 k B T. Thee reult bring u to a very important concluion that a protein cannot find it ite in biologically relevant time if the roughne of the pecific binding landcape i *2 k B T. Although an optimal one-dimenional/three-dimenional combination can peed up the earch, it cannot overcome the lowdown of one-dimenional diffuion. Only fairly mooth landcape ( ; 1 k B T) can be effectively navigated by protein. Speed veru tability Wherea rapid earch require fairly mooth landcape ( ;1 k B T), tability of the protein-dna complex, in turn, require a low energy of the target ite (U min, 15 k B T for a genome of 10 6 bp). In Fig. 5 a, we preent the equilibrium probability P b of binding the tronget target ite with energy U min ¼ U 0 (Gerland et al., 2002) a a function of /k B T. In equilibrium, P b equal the fraction of time the protein pend at the target ite, P b ¼ exp½ÿbu 0Š : (19) M + exp½ÿbu i Š i¼0 Since the target ite i not eparated from the ret of the ditribution by a ignificant energy gap, P b i comparable to 1 (which i the natural requirement for a good regulatory ite) only at k B T. Fig. 5 b how the optimal earch time at the correponding value of /k B T. High roughne of k B T required for

7 Kinetic of Protein-DNA Interaction 4027 FIGURE 4 (a) Dependence of the earch time on the nonpecific binding energy. (b) The parameter pace. The dahed line correpond to optimal parameter and E n connected by Eq. 18. tability of the protein-dna complex lead to atronomically large earch time. In contrat, a protein can effectively earch the target ite at, 1 2 k B T. Thi bring u to the central reult that the ability to tranlocate rapidly along the DNA clearly cannot comply with the tability requirement. Requirement of high tability at the target ite, P b ;1 (or P b ;1/N p,ifn p copie of the protein are preent), yield an etimate for the minimal of p ; k B T ffiffiffiffiffiffiffiffiffiffiffiffi 2lnM; 5 k B T; (20) given a genome ize M ¼ From the above analyi, an obviou conflict arie: the ame energy landcape cannot allow for both rapid tranlocation and high tability of tate formed at ite with the lowet energy. Thi conflict i imilar to the peed-tability paradox of protein folding formulated by Gutin et al. (1998): FIGURE 5 (a) Stability on the protein-dna complex on the cognate ite meaured a the fraction of time in the bound tate at equilibrium. (b) Optimal earch time a a function of the binding profile roughne, for the range of parameter 10 ÿ4 # t 3d # 10 ÿ2, 10 ÿ10 # t 0 # 10 ÿ6. rapid earch in conformation pace require a mooth energy landcape, but then the native tate i untable. In protein folding, thi conflict i reolved by the preence of a large energy gap between the native tate and the ret of the conformation (Finkeltein and Ptityn, 2002; Pande et al., 2000). A evident from Fig. 1, no uch energy gap eparate cognate ite from the bulk of other (random) ite. In fact, the energy function in the form of Eq. 5 cannot, in principle, provide a ignificant energy gap. Increaing the number of TF cannot reolve the paradox either (ee Appendice D and E). An alternative olution mut be ought. The two-mode model The earch-peed tability paradox ha already been qualitatively anticipated by Winter et al. (1989), who therefore concluded that a conformational change of ome ort

8 4028 Slutky and Mirny hould exit that would allow fat witching between the pecific and the nonpecific mode of binding. In the nonpecific mode, the protein i liding over an eentially equipotential urface (in our term, non-pec ¼ 0), wherea ite-binding take place in the pecific mode ( pec k B T). A protein in the nonpecific binding mode i unaware of the DNA equence it i bound to. Thu, it hould permanently alternate between the binding mode, probing the underlying ite for pecificity. Thi model naturally raie a quetion about the nature of the conformation change. Originally, it wa decribed a a microcopic binding of the protein to the DNA accompanied by water and ion extruion. However, numerou calorimetry meaurement and calculation (Spolar and Record, 1994) how that uch a tranition i uually accompanied by a large heat capacity change DC. Thi DC cannot be accounted for, unle additional degree of freedom, namely, protein folding, are taken into account. On-ite folding of the trancription factor may involve ignificant tructural change (Flyvbjerg et al., 2002; Bruinma, 2002; Kalodimo et al., 2004) and take a time of ;10 ÿ4 10 ÿ6 (Akke, 2002) (compared to a characteritic on-ite time of t 0 ;10 ÿ7 10 ÿ8 ). We conclude that conformational tranition between the two mode involve (but i not limited to) partial folding of the TF. If the TF i to probe every ite for pecificity in thi fahion, it would take hour to locate the native ite. We note, however, that if there wa a way to probe only a very limited et of ite, i.e., only thoe having high potential for pecificity, the earch time would be dramatically reduced. From the previou ection it i clear that a relatively weak ite-pecific interaction (i.e., mooth landcape, ; k B T) doe not ignificantly affect the diffuive propertie of the DNA and the total earch time. If thi landcape, however, i correlated with the actual pecific binding energy landcape (with ; 5 6 k B T), the pecific ite will be the tronget ite in both mode. The protein conformational change hould occur therefore mainly at thee ite, which contitute trap in the mooth landcape. Since uch ite contitute a very mall fraction of the total number of ite, the tranition between the mode are very rare. We therefore ugget that there are two mode of protein- DNA binding: the earch mode and the recognition mode (Fig. 6). In the earch mode, the protein conformation i uch that it allow only a relatively weak ite-pecific interaction ( ; k B T) (Fig. 6, top). In the recognition mode, the protein i in it final conformation and interact very trongly ( r $ 5 k B T) with the DNA (Fig. 6, bottom). If two energy profile are trongly correlated, then the lowet-lying energy level (i.e., trap) in the earch mode (# ÿ5 k B T) are likely to correpond to the tronget ite in the recognition mode (putatively, the cognate ite). The tranition between the two mode happen mainly when the protein i trapped at a low-energy ite of the earch landcape. In thi fahion, the one-dimenional diffuion coefficient D 1d i ; time maller than the ideal limit, but the earch time in the optimal regime i reduced only by a factor of ;3 10 (Eq. 11). The coupling between the conformational change and aociation at a ite with a low-energy trap i likely to take place through time conditioning. Namely, the folding (or a imilar conformation tranition) occur only if the protein pend ome minimal amount of time bound to a certain ite. Thi tatement i baically equivalent to aying that the free energy barrier that the protein mut overcome to tranform to the final tate mut be comparable to the characteritic energy difference that control hopping to the neighboring ite. The protein conformation in the recognition mode hould be tabilized by additional protein-dna interaction. If thee interaction are unfavorable, the folded tructure i detabilized; the earch conformation i then rapidly retored and the diffuion proceed a before. If the new interaction are favorable, however, the folded tructure i table and the protein i trapped at the ite for a very long time. For thi mechanim to work, tranition between the two mode of earch ha to be aociated with a ignificant change in the free energy (;5..10 k B T) of the protein-dna complex (ee Fig. 6 c). Such an energy difference between the two tate i required to make the majority of the highenergy ite in the recognition mode le favorable than in the earch mode. A protein would rather (partially) unfold than bind an unfavorable ite. A a reult, ite that lay higher in energy than a certain cutoff exhibit a imilar nonpecific binding energy (i.e., there i a witch into the earch mode of binding). The folding of partially diordered protein loop or helice can provide the required free energy difference between the two mode. Efficiency of the propoed earch-and-fold mechanim depend on the energy difference between the two mode, correlation between the energy profile, and the barrier between the two tate. The barrier determine the rate of partial folding-unfolding tranition. If the barrier i too low, then the protein equilibrate while on a ingle ite, having no effect on earch kinetic. On the contrary, too high a barrier can lead to rear folding event and the cognate ite can be mied. It can be hown that having a barrier of proper ize provide for an efficient earch and table protein-dna complexe. Alternatively, the cognate ite can lower the barrier by tabilizing the tranition tate (i.e., the folding nucleu; ee Abkevich et al., 1994; Mirny and Shakhnovich, 2001), whereby it act a a catalyt of partial folding. (Quantitative analyi of thee factor i beyond the cope of thi tudy, and will be publihed elewhere.) DISCUSSION Specificity for free: kinetic veru thermodynamic The propoed mechanim of pecific ite location i akin to kinetic proofreading (Hopfield, 1974), which i a very general

9 Kinetic of Protein-DNA Interaction 4029 FIGURE 6 Cartoon demontrating the two-mode earch-and-fold mechanim. (Top) Search mode; (bottom) recognition mode. (a) Two conformation of the protein bound to DNA: partially unfolded (top) and fully folded (bottom). (b) The binding energy landcape experienced by the protein in the correponding conformation. (c) The pectrum of the binding energy determining tability of the protein in the correponding conformation. concept for a broad cla of high-pecificity biochemical reaction. The required pecificity i achieved in kinetic proofreading through formation of an intermediate metatable complex that pave the way for irreverible enzymatic reaction. If the reaction i much lower than the lifetime of the complex, then ubtrate that pend enough time in the complex are ubject to the enzymatic reaction, wherea ubtrate that form hort-lived complexe are releaed back to the olvent before the reaction take place. In other word, the ubtrate are elected by kinetic partitioning. In contrat to kinetic proofreading that increae equilibrium pecificity for the price of energy conumption, the earch-and-fold model doe not require any additional ource of energy. The two-mode earch-and-fold model provide a fater on-rate of binding while keeping the equilibrium binding contant unchanged. Naturally, the off-rate i increaed a well. Thi make our two-mode model thermodynamically neutral. Coupling of folding and binding in molecular recognition Several DNA- and ligand-binding protein are known to have partially unfolded (diordered) tructure in the unbound tate. The untructured region fold upon binding to the target. Doe binding-induced folding provide any biological advantage? The idea of coupling between local folding and ite binding ha been around for ome time and wa recently reaeed in the much broader context of intrinically untructured protein (Wright and Dyon, 1999; Dyon and Wright, 2002; Uverky, 2002). Induced folding of thee protein can have everal biological advantage. Firt, flexible untructured domain have an intrinic platicity that allow them to accommodate target/ligand of variou ize and hape; and econd, free energy of binding i required for compentation for the entropic cot of ordering of the untructured region. A poor ligand that doe not provide enough binding free energy cannot induce folding and, hence, cannot form a table complex. William et al. (2001) have uggeted that untructured domain can be the reult of evolutionary election that act on the bound (tructured) conformation, while ignoring the unbound (untructured) conformation. Partial unfolding can alo increae protein radiu of gyration and, hence, increae the binding rate (Shoemaker et al., 2000; Levy et al., 2004). Here we propoe a mechanim that ugget the role of induced folding in providing rapid and pecific binding. Induced folding (or any ort of two-tate conformational tranition) allow a protein to earch and recognize DNA in two different conformation providing rapid binding to the target ite. Importantly, thi mechanim reconcile rapid earch for the target ite with a table bound complex (ee above). The rate of induced folding can alo play a role in determining the pecificity of recognition (M. Slutky and L.A. Mirny, unpublihed). Structural and thermodynamic data argue in favor of ditinct protein conformation for earch along noncognate DNA and for recognition of the target ite. Protein uch a lci, EcoRV, and GCN4 apparently do not fold their untructured region while bound to noncognate DNA (Winkler et al., 1993; Clarke et al., 1991; O Neil et al., 1990); thi upport our hypothei. Heat capacity meaurement on a vat variety of protein- DNA complexe report a large negative heat capacity change in ite-pecific recognition, which i a clear indication of a phae tranition. Thee meaurement upplemented by x-ray crytallography and NMR tructural data were interpreted by Spolar and Record (1994), mainly in term of hydrophobic and conformational contribution to entropy.

10 4030 Slutky and Mirny Thu, folding-binding coupling i now conidered a welletablihed effect for a large et of trancription factor. However, real-time kinetic meaurement were not performed until recently, o that the quetion of the actual mechanim wa left open. Seriou advance in thi direction were made by Kalodimo et al. (2001, 2002, 2004), who oberved a two-tep ite recognition by dimeric Lac repreor. The H/D-exchange NMR data unambiguouly demontrate ite preelection by a-helice bound in the major groove followed by folding of hinge helice that bind to the minor groove element and complete the pecific ite recognition. Although the experiment in thi field were performed with a ingle model ytem, their implication are likely to have a general character. It hould be mentioned that no tranition of thi kind i oberved when the protein i unbound from DNA. A poible reaon for thi can be a ignificant reduction of the free energy barrier for folding, entropic in eence, that accompanie protein-dna aociation. Entropy barrier reduction i a natural conequence of relative anchoring of the variou part of the protein on the DNA caffold. Thermal fluctuation that the aociated protein i ubject to are generally of the order of ;k B T, and their main effect i protein tranlocation along the DNA. From the above analyi, it follow that the tranlocation actually take place only if the protein encounter barrier of ; k B T on it way. In a large enough collection pffiffiffiffiffiffiffiffiffiffiffiffi of ite (M 1), however, potential well of depth ; 2lnM will be preent. If the well depth i larger than the folding barrier height, the probability of on-ite (in-well) folding increae, leading eventually to a table complex formation. (More detailed computational analyi of coupling between folding and binding will be publihed elewhere.) Biological implication The mechanim of three-dimenional/one-dimenional earch decribed above ha everal biological implication. The tudied model, a with any quantitative model, i, of coure, a gro implification of protein-dna recognition in vivo. Depite thi implification, propoed mechanim can be generalized to decribe the in vivo binding. Here we briefly dicu ome of the biological implication of our model. Simultaneou earch by everal protein If everal TF are earching for it ite on the DNA, the total earch time i given by Eq. 15 and i obviouly horter than the time for a ingle TF. For example, if 100 copie of a TF are earching in parallel for the cognate ite, then auming kon cytoplam 10 8 M ÿ1 ÿ1 and a cell of 1 mm 3 volume, we obtain the earch time of t 0.1. Increaing the number of TF molecule can further decreae the earch time, but can have harmful effect due to molecular crowding in the cell. Note, however, that increaing the number of TF molecule to per cell cannot reolve the peed-tability paradox (ee Fig. 5). Search inide a cell: molecular crowding on DNA and chromatin Above we aumed that a TF i free to lide along the DNA. The in vivo picture i complicated by other protein and protein complexe (nucleoome, polymerae, etc.) that are bound to DNA, preventing a TF from liding freely along the DNA. What are the effect of uch molecular crowding on the earch time? Our model ugget that molecular crowding on DNA will have little effect on the earch time if certain condition are atified. Obviouly, the cognate hall not be creened by other DNA-bound molecule/nucleoome. DNA-bound molecule can interfere with the earch proce by hortening region of DNA canned on each round of one-dimenional diffuion. If, however, the ditance between DNA-bound molecule/nucleoome in the vicinity of the cognate ite i greater than n opt ;300 ÿ 500 bp (ee Eq. 13 and Kim et al., 1987), then obtacle on the DNA do not horten the round of one-dimenional diffuion and, hence, do not low down the earch proce. Our analyi alo ugget that equetration of part of genomic DNA by nucleoome can even peed up the earch proce. If DNA-bound protein are eparated by bp, E. coli genomic DNA can accommodate bp/300 bp protein. In other word, all 150 known and predicted E. coli TF can be imultaneouly preent in 100 copie each, and earch for their cognate ite without affecting one other (in fact, they can be preent in 200 copie each, ince optimal earch require 50% of protein to be in olution at any one time). On the other hand, a hort ;50-bp linker between nucleoome in eukaryotic chromatin can increae the earch time ;10-fold. Detail of thi analyi will be publihed elewhere. Funnel, local organization of ite Several known bacterial and eukaryotic ite tend to cluter together. One may ugget that uch clutering or other local arrangement of the ite can create a funnel in the binding energy landcape, which lead to a more rapid binding of cognate ite. Our model ugget that even if uch funnel do exit, they would not ignificantly peed up the earch proce. The propoed earch mechanim involve ;M=n opt ; 10 4 round of one-dimenional/three-dimenional diffuion. So a TF pend all the earch time far from the cognate ite. Only the lat round (out of 10 4 ) will be ped up by the funnel, leading to no ignificant decreae of the earch time. Local organization of ite and other equence-dependent propertie of the DNA tructure (flexibility of AT-rich region, DNA curvature on poly-a track, etc.) may influence preferred localization of TF and lead to fater on-/off-binding

11 Kinetic of Protein-DNA Interaction 4031 rate and fat equilibration on neighboring ite (ee Slutky et al., 2004, for detail). Protein hopping: interegment tranfer Our model aumed that round of one-dimenional diffuion are eparated by period of three-dimenional diffuion. Interegment tranfer i another mechanim that can be involved. If two egment of DNA come cloe to each other, a TF liding along one egment can hop to another. The benefit of thi mechanim i that it ignificantly horten the tranfer time, t 3D. Several example of experimental evidence ugget that tetrameric LacI, which ha two DNAbinding ite, travel along DNA through one-dimenional diffuion and interegment tranfer. We did not conider thi mechanim becaue of the two following conideration. Firt, it i unclear whether TF that have only one binding ite can perform interegment tranfer; and econd, for thi mechanim to work, ditant egment of DNA need to come cloe to each other. Although DNA packed into a cell/nuclear volume croe itelf every ;500 bp, DNA in olution, at in vitro concentration, i unlikely to have any uch elf-croing. Hence interegment tranfer cannot explain the fater-than-diffuion binding rate oberved in vitro. Thi mechanim, however, may play a role in vivo, epecially for protein that have multiple DNA-binding ite. Propoed experiment Our reult propoe everal experimentally tetable prediction. Firt, we predict that the maximal rate of binding i achieved when the protein pend half of the time in olution and half liding along the DNA. Thi reult can be readily verified experimentally by meauring the concentration of free protein in olution that contain DNA but no cognate ite. We alo how how the earch time depend on the energy of nonpecific binding, which, in turn, can be controlled by the ionic trength of olution or by engineering protein with tronger or weaker nonpecific binding. In vivo obervation of the 50/50 rule would ugget that protein are optimized by evolution for rapid earch. Second, we how how the binding rate depend on the average travel time between two random egment of DNA, t 3d. Thi time meaurement (t 3d ) depend on the DNA concentration and the domain organization of DNA. By changing DNA concentration and/or DNA tretching in a ingle molecule experiment, one can alter t 3d and thu tudy the role of DNA packing on the rate of binding. Thi effect ha implication for DNA recognition in vivo, where DNA i organized into domain. Similarly, one can experimentally meaure and compare the binding rate, in the preence of other DNA-binding protein or nucleoome, with analytical prediction. Single molecule experiment and AFM/SFM imaging allow direct obervation of protein trajectory and meaurement of the one-dimenional diffuion coefficient, D 1d,on noncognate DNA. Our formalim, in turn, allow u to calculate the pectrum of pecific binding energy, given D 1d. Such meaurement can be direct tet of our conjecture that one-dimenional earch along noncognate DNA proceed along a moother energy profile. Third, uing protein engineering one can tabilize untructured region of DNA-binding protein (e.g., lci, EcoRV, and GCN4), and tudy the binding rate of thee engineered, rigid protein. Such experiment can tet the propoed earch-and-fold mechanim and hed light on the role of untructured region in determining tability, pecificity, and binding rate. We alo ugget that protein bound to noncognate DNA are not fully ordered. Unfortunately very few tudie (Kalodimo et al., 2001, 2002, 2004) have addreed the mechanim of binding to noncognate DNA. More tudie of tructure, thermodynamic, and dynamic of protein bound to noncognate DNA will deepen our undertanding of pecific protein-dna recognition. CONCLUSIONS We have developed a quantitative model of protein-dna interaction that provide an inight into the mechanim of fat target ite location. We found the range of parameter (pecific and nonpecific binding energie) that are crucial for fat earch and, hence, the robut functioning of gene trancription. Paradoxically, realitic energy cannot provide both rapid earche and trong binding of a rigid protein. Thi allowed u to formulate the peed-tability paradox of protein-dna recognition (which i imilar to the famou Levinthal paradox of protein folding). To reolve the paradox, we propoed a earch-and-fold mechanim that involve the coupling of protein binding and protein folding. The propoed mechanim ha everal important biological implication in explaining how a protein can find it ite on DNA, in vivo, in the preence of other protein and nucleoome and by a imultaneou earch of everal protein. Our model provide, for the firt time, a quantitative framework for analyi of the kinetic of trancription factor binding and, hence, gene expreion. Importantly, our model link molecular propertie of trancription factor to the timing of trancription activation. Proper undertanding of the entire mechanim will hardly be poible without further experimental effort in thee direction. APPENDIX A: DIFFUSIVE PROPERTIES OF THE DNA The derivation conit of two tep. Firt, we decribe the random walk along the DNA in term of number of tep. Next, we calculate the mean time between ucceive tep in the random energetic landcape that provide the timecale for the problem. Such a decoupling, trictly peaking,