Eur. J. Biochem. 98, 557-566 (1979)

Protein - Nucleic-Acid Reaction Kinetics Theoretical Analysis of the Binding Reaction between DNA and RNA Polymerase Paolo U . GIACOMONI Laboratoire de Pharmacologie Moli.culaire (Laboratoire nc 147 : i t , < r i k au Centre National de la Recherche Scientifique), lnstitut Gustave-Roussy, Villejuif, and Department of Biolog). I ~il\ersityof California at San Diego, La Jolla, California (Received October 30, 1978/February 19, 1979)

This paper presen I:, methods developed in order to analyze experimental results concerning the binding of Eschericliici c d i DNA-dependent RNA polymerase to DNA at high and at low DNA concentrations, using the filter retention assay. The basic hypotheses, under which the mathematical expressions for describing the kinetics of binding are derived, are as follows. (a) At low DNA concentration : equivalence and independence of the specific binding sites; first-order dependence of the binding reaction on both DNA and protein concentration. (b) At high DNA concentration : equivalence and independence of the nonspecific binding sites; no direct transfer or one-dimensional sliding of the protein along the DNA. Comparison between theoretical predictions and experimental results at high DNA concentration will alow one to determine the relative value of the rates of binding of RNA polymerase to different promoters (between 1 and 2 in T5 DNA). Binding experiments performed at low DNA concentration are reported in this paper: these results and the analysis which is reported allow one to determine the value of the rate constant of formation of non-filterable complexes for the system fd DNA (replicative form) . RNA-polymerase (k, = 3.3 x lo8 M-’ s-l in 0.1 M NaCl, 0.01 M MgC12).

The synthesis of RNA by RNA polymerase on a DNA template follows a pathway which consists of three main steps [l].These steps are called initiation, elongation and termination. The initiation itself has been schematically subdivided into five steps: (a) binding of RNA polymerase to a specific site (promoter) on the DNA; (b) formation of a productive complex; (c) binding of the initiating ribonucleoside triphosphate to the productive complex; (d) binding of the second ribonucleoside triphosphate; (e) formation of the first phosphodiester bond and liberation of inorganic pyrophosphate. Steps (c), (d) and (e) have been the object of detailed kinetic studies, recently revised and discussed by Smagovicz and Scheit [2]. Steps (a) and (b) can be described as follows: E+D

I ,!

E-D i7+ [:-D*

(1)

where E and D represent free RNA polymerase and promoter DNA, E-D and E-D* represent respectively Enzymes. DNA-dependent RNA polymerase holoenzyme, nucleosidetriphosphate: RNA nucleotidyltransferase (EC 2.7.7.6) ; restriction endonuclease Hind11 (EC 3.1.4.-).

the non-productive and the productive complexes, k,, k d and k, are respectively the rate constants of binding of RNA polymerase to the promoter, of dissociation of RNA polymerase from the promoter and of formation of the productive complex. The relative strengths of two promoters, defined as the ratio of the number of RNA chains initiated in the same time interval at two different promoters under identical physico-chemical conditions [3] will depend on k,, kd, k, as well as on the rate constants involved in steps (c), (d) and (e) for every other promoter in these particular conditions. Therefore, learning about the values of k,, kd and k, appears to be important. The filter retention assay has been the method of choice for learning about protein - nucleic acid interaction kinetics: it is known that nucleic acids bound to proteins such as RNA polymerases and repressors are retained on nitrocellulose filters [4- 71. Using radioactively labelled nucleic acids, it is possible to measure the formation and the dissociation of these non-filterable complexes as a function of time and to determine the kinetic parameters of their reactions. One bound protein molecule promotes the retention of its ligand nucleic acid molecule on the filter [ 5 ] .

558

When there is one single specific binding site per nucleic acid molecule, the amount of nucleic acid retained by the filter is a direct, though not necessarily an absolute, measure of the amount of protein (having both template-binding and filter-retaining activity) bound to the nucleic acid at time t in a non-filterable complex. The characteristic for a protein to have both templatebinding and filter-retaining activity is pointed out here in order to overcome difficulties in defining and measuring the filter retention efficiency. When there are n ( n 2 2) specific binding sites per nucleic acid molecule, the amount of protein (having both template-binding and filter-retaining activity) bound per nucleic acid molecule has to be calculated from the fraction of the labelled nucleic acid retained on the filter according to the binomial distribution law or by the simpler Poisson distribution law as described in Materials and Methods. This paper deals with the analysis of the kinetics of formation of nonfilterable protein - nucleic acids complexes when there is more than one specific binding site per nucleic acid molecule. The analysis of the dissociation kinetics of such complexes has already been reported [8]. Among the questions which can be answered by studying the kinetics of formation of non-filterable complexes, one finds the following. Is the formation of a non-filterable complex a concentration-dependent, second-order reaction? An affirmative answer to this question either implies that the step governed by k , is not rate limiting, or rules out the assumption that only the productive complexes E-D* are non-filterable because, according to Eqn (1) the formation of E-D* from E-D should follow first-order kinetics. Is the formation of a non-filterable DNA . RNA-polymerase complex a diffusion-controlled reaction? Let us recall that for diffusion-controlled reactions the rate constant of association is given by [9]:

where D, and D, are the diffusion coefficients of enzyme and DNA, NA is Avogadro's number and a is the reaction radius (i.e. the sum of the radius of the active site of the protein, as far as DNA-binding 'activity' is concerned, and the radius of its ligand nucleic acid). An affirmative answer to this question would rule out the possibility that different net electrical charges carried by different promoters are effective in biasing the formation of the complex E-D. Another consequence could be that geometrical properties, such as the frequency of breathing [lo], can aflect the value of the mean radius of a promoter and thus the value of k , for every promoter. Several experiments can be designed in order to provide answers to these questions. The algorithms presented in this paper allow one to predict the outcome of DNA . RNA-polymerase binding expeti-

DNA . RNA-Polymerase Binding Kinetics

ments under the hypotheses that the binding sites have identical k , values and that the reaction is first order in both reactants, provided that the number of specific sites is known. Discrepancies between theoretical predictions and experimental results .will allow one to learn about the relative value of the k, for different promoters in the same set. MATERIALS A N D METHODS DNA-dependent RNA polymerase holoenzyme was purified from Eschuichiu coli as previously described [ll]. Enzyme concentration was determined using the absorption coefficient A i z = 6.2 [12]. The replicative form of 14C-labelled fd DNA was purified as previously described and had a specific activity of 9800 counts min-' pg-' [13]. This DNA was incubated with restriction enzyme HindII (gift from H. Kopecka) in order to obtain full-length linear DNA, which was then phenol extracted and dialyzed against the appropriate buffer. Analytical sedimentation analysis and electrophoresis on agarose gels [I1] confirmed the integrity of the linear DNA molecules after incubation with HindII. The DNA . RNA-polymerase associaton kinetics were followed by the filter assay using Schleicher and Schull nitrocellulose filters (BA 85) pretreated according to Lin and Riggs [14]. Binding experiments were performed in binding buffer, which contained 0.04 M Tris-HC1 pH 7.9, 0.01 M MgC12, 0.1 M dithiothreitol and 0.05 M, 0.1 M or 0.15 M NaC1, as indicated. The concentrated enzyme solution was carefully diluted on ice in binding buffer to a concentration of 0.4 mg ml- * . 2 pl of this solution were carefully layered without bubbling on the bottom of a small beaker and 1-2.5 pg labelled DNA in 5 ml binding buffer at 37 "C was immediately ( f = 0) poured into the same beaker. The reaction took place at 32-34°C. 1-ml aliquots were withdrawn at the indicated times and filtered. The filters were dried and counted. Filtration rate was 0.5 ml s-'. In the absence of enzyme 5.5% of the radioactivity was retained on the filters. This value was subtracted from the radioactivity retained on the filters during the association kinetics experiments and the value of r , the mean number of protein molecules bound per nucleic acid was determined according to the Poisson distribution law, as follows : let q be the fraction of nucleic acid retained on the filter: if r 6 n the fraction of nucleic acid free of bound protein is p = 1 - q = exp (- r) and r = - In (I - q). When q is small ( q 5 0.2) then q % r . It can be easily verified that the Poisson distribution approximation is correct within less than 3 (as compared to the binomial distribution) if n 2 6 and Y < 1. The total DNA concentration, [DNA],). varied between 0.05 and 0.125 nM (molecules of DNA)

(x

5 59

P. U. Giacomoni

assuming a molecular weight of 4 x lo6 for the replicative form of fd DNA. An enzyme preparation does not contain, generally, 100 % active molecules. Moreover, upon dilution, the fraction of molecules having DNA-binding and filterretaining activity can also be reduced. Thus the total concentration of 'active' enzyme, [E]o, was experimentally determined from the value of r after the completion of the binding reaction, i.e. after a plateau is reached. In fact it will be shown in next section that Y (a) = [Elo/[DNAIo and, since [DNA10 is known and r ( c 0 ) is measured, [El0 is easily determined. In order to determine the value of k,, experimental data are compared with the analytical expression of r = r ( t ) , which will be derived in next section [Eqn (9)]. This equation contains three parameters : [Elo, n and k,. The value of [El0 is determined as beins [El0 = r (cc) [DNA10 and the replicative form of fd DNA has six strong binding sites [ l l , 151 so n = 6 . It is thus possible to vary the value of k, in Eqn (9) in order to fit the experimental data: the value of k, yielding the best fit is considered the value of the rate constant of association for the DNA . RNA-polymerase binding reaction in the particular experimental conditions.

RESULTS Binding experiments of RNA polymerase to DNA are generally performed in the presence of NaCl and of MgC12. De Haseth et al. [16] have shown that the equilibrium constant of the complex RNA-polymerase . non-promoter-DNA range5 from 5 x lo6 M in 0.04 M NaCl and 0.01 M MgClz to 6 x lo4 M-' in 0.16 M NaCl and 0.01 M MgC12. The equilibrium constant for the specific DNA . RNA-polymerase complexes has been estimated for T7 and fd DNA and ranges from 4 x 10'l M-' in 0.12 M NaCl and 0.01 M MgC12 to 2 x 10'' M - ' in 0.05 M NaCl and 0.01 M MgCl2 [S, 131. When the binding equilibrium between two reactants is considered, one recalls that at a free concentration of one of the reactants equal to the reciprocal of the equilibrium constant, half of the possible complexes with the other are formed. At lower concentrations the equilibrium is displaced towards dissociation, while at higher concentrations the equilibrium is displaced towards association. In order to avoid non-specific binding in 0.1 M NaC1, 0.01 M MgCl2 one should perform experiments at a non-specific site concentration less than 1 pM, which corresponds to 0.3 pg ml-' : in a long DNA molecule (in case of low occupancy by ligand proteins) there are approximately as many potential non-specific sites as there are base-pairs [ 171. Therefore nucleotide concentrations above about 10 pM ( 3 pg m1-l) will favor the

formation of the non-specific complexes and reduce the concentration of free R N A polymerase able to bind directly to the promoters. Further increase in DNA concentration will cause the rapid formation of non-specific complexes, followed by the formation of specific, non-filterable complexes as discussed later in this section. In this paper, for DNA concentrations, low and high will mean well below and well above the reciprocal of the equilibrium constant for non-specific binding. Formation of Non-filterable Complexes at Low D N A Concentration Schaller and coworkers [3] have observed secondorder kinetics of binding of R NA polymerase to certain of the fd DNA restriction fragments. This observation allows one to derive a mathematical expression for the kinetics of formation of non-filterable complexes under a set of theoretical assumptions. When one considers the binding of a protein to specific sites on DNA in a solution at low concentration, one can imagine the proteins diffusing isotropically in the medium, randomly encountering the DNA without being driven by long-range signals and binding to the nucleic acid only after having hit one of the specific sites. This model can be summarized in the following set of hypotheses. a) The specific sites on the nucleic acid are kinetically equivalent, i.e. their rate constant of association to the protein (k,) are equal and different specific sites do not display different net electrical charges nor different sizes able to bias the outcome of a random binding of the proteins to the nucleic acid. b) There is no correlation between the kinetic path leading to the association of a protein at a given site and its association at another site of the same nucleic acid molecule, i.e. there is neither one-dimensional sliding of the protein along the nucleic acid nor direct transfer of the protein from site to site. c) The specific binding and the formation of nonfilterable complexes do not require conformational (or other) changes constituting a rate-limiting step. d) The binding reaction is first order both in protein and in nucleic acid concentration. At this point of the analysis there is no justification for these theoretical assumptions in the case of the binding of RNA polymerase to promoters; for instance, hypothesis (b) would not be justified for the binding of lac repressor to lac operator [4,18,19]. The rationale of this procedure is that from these hypotheses one can obtain a mathematical expression describing the kinetics of binding : comparison between theoretical predictions and experimental results will allow one to determine the value of the rate constant of association, k,, which could be used with

560

D N A . RNA-Polymerase Binding Kinetics

Eqn (2) to obtain a rough estimate of the value of the reaction radius, i.e. the sum of the radius of the active site on the enzyme plus the radius of the sphere having equivalent cross-section as the promoter. (This value is otherwise difficult to evaluate, but can be guessed to range between 2 and 5 nm: the promoter can be visualized as a cylinder 40 base-pairs long [20] the diameter of which is close to 2 nm, and the active site on the RNA polymerase can be visualized as a cylinder of approximately the same size.) An unexpectedly large value for the reaction radius would suggest that hypotheses (a) and (b) are not valid. An unexpectedly small value for the reaction radius would suggest either that hypothesis (c) does not apply or that the formation of non-filterable complexes can only take place when the two interacting molecules happen to be well oriented relatively to one another. Let us now proceed to the derivation of a mathematical expression for the DNA . protein binding kinetics, under the hypotheses (a) to (d). As a matter of fact, when the concentration of non-specific sites is low, the binding of a protein E to a specific site S on a nucleic acid molecule can be represented as follows :

which means that the initial slope of the function r = r(t) for a second-order reaction is independent of the nucleic acid concentration, and does only depend on the total protein concentration and on the number of specific binding sites on the nucleic acid molecule. Eqn ( 5 ) can be rearranged to yield:

Integration of Eqn (7) leads to

and finally,

1 - exp { - (n [DNAIo-

A,

E+S$F-S. hd

When sufficiently far from equilibrium, i.e. for t 4 l/kd, one can write:

Eqn (9) indicates, that r If [El0 < n [DNAIo, then

!iPI (3) where [E-S], [El and [S] represent respectively the molar concentrations of the complex, and of free proteins and specific sites at time t. Let [DNA10 be the total concentration of nucleic acid, with n specific sites per molecule having identical k, values. Let [El0 be the total protein concentration, and r = r(t) be the mean number of bound proteins per nucleic acid molecule at time t. It follows that n [DNA10 is the total concentration of binding sites; [E-S] = r(t)[DNA]o is the concentration of complexed binding sites at time t and [S] = {n - r(t)}[DNA]o is the concentration of free binding sites. From Eqn (1) one obtains dr(t) [DNA10 = k, (n - r) [DNA10 [El . dt

(4)

Since [El = [E]o - r [DNAIo, it follows ~

dr = k , (n - r)[EIo - k, (n - 7) r [DNA10 . dt

From Eqn ( 5 ) it is easily seen that

(5)

. 1

=

r(t) tends to level off.

r(t) = [Elo,"DNAlo

(10)

allowing the concentration of the enzyme molecules having both DNA-binding and filter-retaining activity to be estimated easily. Moreover, if the value of n is known, Eqn (9) allows one to calculate r(t) for different values of k , and to determine the value of k, yielding the best fit with the experimental data, if the reaction is first order both in DNA and enzyme concentration. As an experimental example for illustrating the use of Eqn (9), the kinetics of binding of RNA polymerase to fd DNA are reported in Fig. 1. The experimental value of the mean number of enzyme molecules per DNA molecule, r, is plotted versus time t. The best fitting theoretical values of r = r(t) are also reported. The values of k , giving the best fit vary slightly with ionic strength. This is in agreement with what one expects from a diffusion-controlled reaction [19]. Moreover, the value of k , from curve 3 in Fig. 1 is k, = 3 x lo8 M-' s-'. This value has been obtained using Eqn. (9) assuming n = 6. Since the determination of the value of r = r(t) depends on the product aka, it is unlikely that the value of k , so determined is largely misestimated because the value of n is known within less than a factor of two [ l l , 151. Seeburg and coworkers [3] found that fd DNA (replicative form) and RNA polymerase associate following a secondorder reaction, and they measured k, as being k , = lo7 M - ' s-' in 0.12 M NaCl, that is ten times

P. U . Giacomoni

561

Formation of Non-jilterable Complexes at High DNA Concentration

cc 0

I 5

10

15

23

25

Time

30

35

40

45

50

(5)

Fig. 1. Kinetics of association of,fZ D N A (replicative fbrm) with E. coli R N A polymerase. (A) Binding buffer contained 0.05 M NaCl; (B) binding buffer contained 0.1 M NaCl; (C) binding buffer contained 0.15 M NaCI. Curves 1, 3, 5 (0) D N A concentration was 0.05 nM. Curves 2, 4, 6 (0)D N A concentration was 0.125 nM. The solid lines show the best-fitting theoretical determination of r = r ( t ) according to Eqn (9). Best fit was obtained using the following values of k,: curve 1 : k, = 4.5 x 10' M - ' s-" , curve 2 : k, = 2 x 10' M-' s-' ; curve 3: k, = 3 x lo* M - ' s - l . , curve 4 : k , = 1 x 10' M - ' S - ' ; c u r v e 5 : k a = 2 x 1 0 s M - ' s - ' and curve 6 : k , = 1 x 10' M - ' S - '

smaller than the values reported in this paper. A possible explanation is that these authors did not check the fraction of their enzyme preparation conserving both DNA-binding and filter-retaining activity : in Fig.4 of their paper, upon addition of one enzyme molecule per D N A molecule, it can be seen that 35 % of supercoiled DNA and 5 % of relaxed DNA are retained on nitrocellulose filters. Since these authors mixed 0.01 pmol supercoiled DNA with 0.07 pmol relaxed DNA, the fraction of total DNA retained on the filter is (0.35 x 0.01 + 0.05 x 0.07)/0.08 = 0.087 instead of 0.63, as should be expected according to the Poisson distribution law. If one bound enzyme promotes the retention of DNA on filters [5] then this figure of 9 % of DNA retention on filters upon addition of one enzyme molecule per DNA molecule corresponds to an actual imput of 0.1 'binding active' enzyme molecule per DNA molecule. Since these authors measured the rate of complex formation at different RNA polymerase concentrations, if a similar enzyme preparation has been used, the discrepancy between their determination of k, and the one reported in this paper is easily understood.

A typical binding experiment performed at high DNA concentration is to mix RNA polymerase with restriction fragments of a radioactive DNA, incubate for a determined time, stop the binding reaction upon addition of non-radioactive DNA in excess, filter, elute and analyze by gel electrophoresis the amount of each fragment which has been retained on the filter [3,21]. From these experiments it appeared that the relative amount of fragments retained on the filters by E. coli RNA polymerase varied by as much as factor of 10. As I indicate below these data are nevertheless consistent with a model in which the relative values of the rate constant of association to RNA polymerase for the binding sites in HpaII fd DNA and Hind111 T5 DNA vary only within a factor of two, using the following analysis. This analysis, of course, only puts forward an alternative to the interpretation offered by the authors of references [3] and [21]. Let us mix RNA polymerase with a population of identical DNA molecules, A4 base pairs long and carrying n identical specific binding sites, at high DNA concentration. One can imagine that each individual protein molecule undergoes an apparently disordered process of binding and dissociating to and from nonspecific sites until a long-lasting non-filterable complex is formed. Such an image could be very difficult to translate in mathematical terms. It is nevertheless possible to treat this phenomenon mathematically under the following hypotheses. a) Every DNA fragment carries A4 identical and independent non-specific sites. b) The fate of a polymerase bound to a non-specific site is the dissociation from that site after an average time 7,: this is equivalent to the assumption that the binding to a promoter does not take place via onedimensional sliding along the DNA, nor by direct transfer of an RNA polymerase molecule initially bound to a non-specific site. c) The reaction can be regarded as a succession of trials. Every trial except the first begins with all the proteins bound (specifically or non-specifically) to the DNA fragments, distributed among them according to the law of Poisson. After a time z, (corresponding to the life-time of the non-specific complexes) all the non-specifically bound proteins dissociate from the DNA and redistribute themselves among the DNA fragments again according to the law of Poisson. This redistribution takes place in a time d t , which corresponds to the average time-of-flight of a protein from a site to another site. Thus a trial takes place in a time interval equal to z, At. Let r be the mean number of enzyme molecules per DNA molecule: the DNA can be subdivided into

+

DNA . RNA-Polymerase Binding Kiiictics

562

subpopulations of molecules carrying zero, one, two or more enzyme molecules. The fraction of DNA molecules with 1 RNA polymerase non-specifically bound to it will be rl Fl = - e P r (12) I! and r = r ( t ) varies from trial to trial. According to the hypotheses previously stated, at time t = 0 an enzyme molecule-interacting with a DNA fragment will have a probability P of binding to a specific sites thus forming a non-filterable complex having a mean life z much longer than the time necessary for the binding experiment to be carried out [5,8,13,21] and a probability Q = 1 - P of forming a non-specific complex having a short mean life 7., If a non-specific complex is formed, after an average time zn the RNA polymerase molecule will dissociate and, on rebinding to a DNA molecule will attach to a specific site with the same probability P. This process will be repeated until a long-lasting non-filterable complex is formed. This is clearly a Bernoulli trial. Therefore the probability p ( T ) of an RNA polymerase molecule forming a non-filterable complex at the Tth trial is p(T) = Per-'

(13)

so that the fraction of enzyme molecules specifically bound to a specific site after T trials will be P)" dm.(14) So far we have considered the probability for an enzyme to bind to a specific site. Experimentally one generally measures the amount of DNA which is retained on the filter (and one specifically bound RNA polymerase promotes the retention). In order to calculate the fraction of the DNA population which will be retained on the filter after T trials, one has to consider separately the subpopulations having one, two, three or more enzyme molecules to deal with and assume that, if there are 1enzyme molecules, the probability of formation of a nonfilterable complex will be IP, and the probability of formation of a non-specific complex will be 1 - IP. Of course, for a determined DNA fragment, the value of I can change from trial to trial, but the number of DNA fragments interacting with 1 enzyme molecules remains practically constant throughout the reaction although the proteins specifically bound to a DNA molecule have to be considered as withdrawn from the trial, so that the value of r = r ( m ) is actually decreasing as the number of trials m increases. Eqn (12) has to be re-written as

and, iff* is the fraction of RNA polymerase specifically bound to DNA after m trials one has:

where ro is the protein to DNA ratio at time t = 0 and f * can be calculated using Eqn (14). Thus the fractionf(T) of DNA molecules having formed a nonfilterable complex after T trials will be

Nevertheless elementary numerical analysis on Eqns (12) and (12.2) shows that for every value of r , in the case I = 1, as long asf*(m) remains below 0.4 the discrepancy between Fl [ r ( m ) ] and FI [r(O)] remains within 11 % and it becomes greater than 30% only when f * assumes values above 0.6. The discrepancy is greater for greater values of 1 (for 1 = 2 the discrepancy is of the order of 20% forf* = 0.2) but this will not influence much the results of the calculation if one approximates Eqn (15) with

IP)"

(16)

or even by T

f ( T ) = Fl [r(O)]mC =l P ( l -P)"

(1 7)

which can be written as I' [(l - P)' - I ] f ( T ) = F I ]P (1 - P)" dm = F l p ____ -1% In (1 - P ) (18)

because, at low values of r, one can show that FI vanishes for 12 2: when r = 0.3, F1 (0.3) = 0.22, FZ(0.3) = 0.03 and F3 (0.3) = 0.003 and, when r = 0.1, Fl (0.1) = 0.09, F2 (0.1) = 0.0045 and F3 (0.1) = 0.00015. Since the DNA concentration is such that initially the equilibrium is dramatically displaced towards the nonspecific binding, it is not unreasonable to assume that the probability of forming a specific complex upon hitting a specific site (as the probability of hitting the bulls-eye in a target) is equal to the ratio of the number of specific binding sites to the number of non-specific binding sites in the DNA molecule:

Eqns (15) and (19) can be applied for interpreting the results in reference [21] according to this analysis. The experiments were performed at a polymerase-togenome ratio varying between less than 1 and 2, but T5 DNA was cut by restriction enzyme Hind111 which yields about ten fragments so that the polymerase-to-fragment ratio was well below 1. This fact

P. U . Giacomoni

563

F 0.8

1.0 -

-

F

p 0.8 -

2

0

6

4

10

8

12

14

16

18

20

I

Fig.2. Plot

I

P(I

of

- Pi"

dm versu.7 P at ciiff&wt values

of

T.

Fig.3. Plot o f j P ( I - P)" dm = f(T)/Fl,for the Hindlll TS DNA 0

0

These curves can be used with Eqn (12) in order to evaluate f ( T , P ) in Eqns (16) and (18)

,fragmeiits. The value of the probabilities of binding to the ith fragment, Pi = ni/Mi can be obtained by the values of n, and M , in Table 1

Table 1. Coniparison between predicted and experimental relative rates of binding of R N A polymerase to the fragments of Hindlll-restricted TS DNA In columns M and n are reported the number of base-pairs and of specific sites in every fragment (Bujard, personal communication and 1221). The experimental relative rates of binding are obtained from [21] assuming as unit the rate of binding to fragment A. A is calculated recalling that the standard deviation of the experimental values was 14% of the sample value [21]. The limits of the interval of confidence are obtained adding or subtracting A to and from the experimental value of the relative rate of binding, The predicted relative rates of binding are calculated using Eqn (18) assuming T = 1300. The correction coefficient is the coefficient by which the value of Pi = n i / M , has to be multiplied in order to obtain the predicted relative rate of binding within the interval of confidence ~~

T5 Hind111 fragment

hl

17 200 15 200 13800 12900 11 600 10 700 6 700 6 400 3 900

n

Experimental relative rate of binding

A

1 0.303 1.06 0.636 0.606 1.06 0.393 1.969 1.515

0.085 0.296 0.178 0.169 0.296 0.110 0.55 0.42

allows us to use Eqn (18) for determining the value off( T ) for every fragment considered separately and independently from the other fragments, as if the reaction were taking place in different test-tubes, each containing one type of restriction fragment (at the same molar concentration of fragments) and having equally distributed the RNA polymerase among the test-tubes. It is thus possible to predict the outcome of the experiments of Gabain and Bujard once the time interval between two successive trials is known. It is not necessary to know the value of z, + A t with extreme accuracy because it can be seen in Fig. 2 that the ratiof(P1, 7')If(P2,T )does not change dramatically over a wide range of T. Since it is not unreasonable to assume that the time interval between two successive trials is 0.1 s (see Discussion), a typical experimental value for T would be T = 1200 for the

~~

Interval of confidence

Predicted relative rate of binding

-

1 0.43 0.86 0.716 0.782 1.8 0.8818 1.527 1.33

0.218-0.388 0.764- 1.350 0.458-0.814 0.436-0.775 0.764-1.350 0.283-0.503 1.420-2.520 1.09 - 1.930

~~~~

~

Correction coefficient -

0.90 1 1 0.98 0.6 0.5 1 1

experiments in [21] where the binding reaction takes place in 2 min. It is thus possible to calculate the value of the fraction of each fragment retained on the filter as a function of the number of trials in a first-order approximation [i.e. using Eqn (18)]. The result of such a calculation is plotted as fi(T)/R versus T for every DNA fragment in Fig. 3. The theoretical predictions are compared to the experimental results in Table 1. The experimental indeterminacy A and the interval of confidence for the ratio of the rate of binding of two fragments are calculated, recalling that in [21] it is reported that the overall error amounts to 14% of the sample value. The values of the number of binding sites in every fragment have been obtained by electron microscopy analysis [22] (and Bujard, personal communication).

564

DNA RNA-Polymerase Binding Kinetics

Table 2. Comparison of experimental data and theoretical predictions on the.fruction of D N A retained on filters at time t in the experiment in [23] The calculation has been performed using the first nine terms in Eqn (16), assuming [E]o/[DNAl0 = 4, assuming 3 promoters and 39800 base-pairs on T7 DNA, considering t, = 0.014 s ( K , = 1.4 x lo6 M-’ in 50 mM NaCI, 10 mM MgClz [16]) and assuming a time-of-flight equal to 0.06 s if DNA is at 6 fig ml-’ or equal t o 0.05 s if D N A is at 30 pg ml-’ (see Discussion) so that the time interval between two trials is 0.074 s in the first case and 0.064 s in the second case Time

DNA retained when initial concentration is

no. of trials

experimental value

predicted value

”/,

S

~

10 20 30 40

135 270 405 540

7 15 20 22

3.9 8.0 11.0 14.3

The results of some experiments could well be such that this theory and the use of Eqn. (19) do not provide a good fit. In order to have the predicted values of the relative amount of retained fragments within the experimental interval of confidence, one might have to multiply the probability of binding of RNA polymerase to every fragment by a correction coefficient. These coefficients are listed in the last column in Table 1 and could be used as parameters for characterizing the relative rate of binding of RNA polymerase by different promoters. For the experiment of von Gabain and Bujard, these coefficients vary between 0.5 and 1. A consequence of the assumptions leading to Eqns (15 - 18) is that at high DNA concentrations the rate of binding will depend only on the [E]o/[DNA]o ratio and only slightly on the total value of the concentration of DNA. Experimental evidence supporting this prediction can be found in the literature. In Fig. 1 of [23] it can be seen that with [E]o/[DNA]o equal to 4 and at DNA concentrations between 6 and 30 pg ml-‘, the two binding curves are nearly indistinguishable. Moreover from Eqn (16 ) one calculates what fraction of DNA will be retained on filters at different times in this experiment. The results of this calculation are reported in Table 2. It appears that the theoretical values are slightly inferior to the experimental ones. The agreement between predicted and experimental values can be considered satisfactory if one recalls that in this experiment in [23] the time between dilution and completion of filtration was approximately 10 s, so that the experimental values themselves are approximated by excess. DISCUSSION This paper describes methods developed in order to analyze experimental data concerning the kinetics

no. of trials

expermental value

‘i:

- __

160 320 480 640

_ _ _ _ ~___ 7 14 17

20

predicted value _ _ 4.55

~ __

8.8 13 16.7

of binding of RNA polymerase to DNA, taking advantage of the fact that RNA polymerase and DNA form non-filterable complexes. Assuming that nonfilterable complexes only exist when the enzyme is bound to specific sites on the DNA, it is possible to analyze the formation of these complexes at low and at high DNA concentration, provided that one knows exactly the number of specific sites on the DNA. The analysis of data obtained at high DNA concentrations has been performed under the following hypotheses. a) The distribution of the enzyme among the different restriction fragments is such that the first-order approximation [Eqn (18)] can be used. The experiments in [3] and [21] fulfil this hypothesis. b) The time interval between two successive trials is at least 0.1 s. This hypothesis is justified because at every trial an enzyme molecule detaches from a nonspecific site and will bind again either to the same DNA fragment or to another DNA fragment in the reaction vessel. The probability of rebinding to the same fragment can be expressed as 6Q/4n: where 652 is the solid angle under which the individual RNA polymerase molecule ‘sees’ the DNA fragment. Now, at the DNA concentration used in [21] (1 -2 nM) one can estimate the mean distance between two DNA molecules as s = 1.7 pm. If the square of this value can be used as the mean-square value of the displacement (S2) of the RNA polymerase having a diffusion coefficient D , then, from Einstein’s equation [24] S2 = 6Dt one obtains t = 0.05 s. This value, added to 0.05 s corresponding to the mean life of the non-specific complex at 0.04 M NaCl 0.01 M MgCl2 [16] yields 0.1 s between two trials. The mean life of the nonspecific complex has been estimated as the ratio of the non-specific dissociation constant to the rate constant of association enzyme . non-promoter DNA, which is assumed to be 10’ M-’ S - l .

565

P. U. Ciacomoni

c) The probability of binding to a promoter in the ith fragment is Pi = ni/Mi. This hypothesis is equivalent to the assumption that all promoters have identical k, values. A priori this assumption is not justified: it is indeed the hypothesis which can be tested by the comparison of theoretical predictions with experimental data. From the data in Table 1 it appears that the relative values of the average rate constant of association to RNA polymerase for the promoters in different Hind111 restriction fragments of T5 DNA vary between 1 and 2. As far as the formation of non-filterable complexes between E. coli RNA polymerase and the replicative form of fd DNA is concerned, Seeburg et al. [3] performed experiments similar to those reported in [21] using HpaII-digested fd DNA. A quantitative comparison between theoretical predictions and experimental data is not possible because the values of the relative amount of the retained fragments are not reported in [ 3 ] . A qualitative analysis can be nevertheless carried out. From [3,11,15] it is known that fragment A has 1500 base-pairs and two promoters, fragment B has 850 base-pairs and one promoter, C has 750 base-pairs and one promoter, D has 600 base-pairs and one promoter and fragment H has 380 base-pairs and one promoter. Thus, if the promoters have identical k, values, it follows that PA pB = 12x 10-4, pC = 13.3 x 10-4, = 13.3 x PD = 16 x The mean life and PH = 26 x of the specific complex formed by the enzyme with fragment H is very short [3] so that enzymes can be bound and released several times during the incubation time. A consequence of this fact is that it becomes very difficult to estimate the rate of binding of enzymes to fragment H. For the other promoters, the mean life of the non-filterable complex is longer than the time required for the experiment. Although PC = PA one expects that C will bind RNA polymerase slightly faster than A, because its diffusion coefficient is bigger, and the expected hierarchy of the fragments in the binding reaction is D > C > A > B, in agreement with the experimental results in [3]. When the binding experiments are performed at low enzyme and DNA concentrations, the non-specific equilibrium can be neglected and the binding reaction has been analyzed from the kinetic point of view under the following hypotheses. a) The specific binding sites of fd DNA have identical k , values. This hypothesis, which could lead to underestimation of the maximum value of k, if the promoters have different k , values, is nevertheless justified by the considerations just discussed. b) The reaction is first order both in DNA and in enzyme concentration. This hypothesis is justified by observations reported in [3]. An analytical expression has been derived to express the mean value of enzyme molecules bound per DNA molecule as a function of

time [Eqn (9)]. This equation has been used for determining the value of the rate constant of association of the DNA . RNA-polymerase complex, k,. One could argue that this equation is function of three curvefitting parameters: n, [Elo and ka and that the same equation is used for determining the value of two of them, i.e. [Elo and k,. Now, the value of n has been independently determined [11,15] and it has to be noted that the value of r ( c o ) does not depend on the shape of Eqn (9): it appears evident that, once the equilibrium is reached, the mean number of enzymes bound per DNA molecule cannot be very different from [E]o/[DNA]o.Eqn (10) has been explicitly written to show that Eqn (9) satisfies an experimental boundary condition. It was thus possible to use Eqn (9) through a trialand-error method in order to establish the value of k , ( 3 x lo* M - ' s - l in 0.1 M NaC1, 0.01 M MgC12). This value (and the others obtained at different ionic strengths) is very close to the one expected from the von Smoluchowski equation [9]

+

if one uses D, D, = D, = 0.3 x cm2 s-l and R, = 3 nm, where Re and R, are the radii of the enzyme binding site and of the DNA site. One has to recall that the theoretical value of k, in the von Smoluchowski equation has been obtained under the hypothesis that every encounter between two particles leads to the formation of a complex, provided that one particle hits the other particle on one of the two sides which can be attributed to a particle (this is why the orientation factor is 27c instead of 471). Should this hypothesis not be verified for the DNA . RNA-polymerase system, then the value of the true rate constant of association would be a function of another orientation factor. a = Re

+

I am indebted to Dr Bujard for communicating to me unpublished results and to Tim Lohman for helpful discussion. 1 would like to thank Jean Bernard Le Pecq (Villejuif) and Shella Fuhrman and Barry Chelm (San Diego) for critically reading and commenting on the manuscript. During my staying in San Diego I was a recipient of a Fellowship Grant from the Damon Runyon-Walter Winchell Cancer Fund, and of a NATO fellowship.

REFERENCES 1. Anthony, D. D., Zeszotek, E. & Goldthwait, D. A. (1966) Proc.. N a t l A c a d . Sci. U . S . A . 56, 1026-1033. 2. Smagowicz, J. W. & Scheit, K. H. (1977) Nucleic Acids Rrs. 4 , 3863- 3876. 3. Seeburg, P. H., Nuesslein, Ch. & Schaller, H. (1977) Eur. J . Biochem. 74, 107-113. 4. Riggs, A. D., Bourgeois, S. & Cohn, M . (1970) J . Mol. Biol. 53, 401 -417. 5. Hinkle, D . C. & Chamberlin, M. J. (1972) J . Mol. Biol. 70, 157- 185.

P. U. Giacomoni: DNA . RNA-Polymerase Binding Kinetics

566 6. Pongs, 0. &Ulbrich, N. (1976) Proc. NutlAcud. Sci.U . S . A . 73,

7. 8. 9. 10. 11. 12.

13. 14. 15.

3064- 3067. Ballivet, M., Reichardt, L. F. & Eisen, H. (1977) Eur. J . Biochem. 73, 601 -606. Giacomoni, P. U. (1976) FEBS L.crt. 7-7. 83-86. von Smoluchowski (1917) Z . Phjs. Chem. 92, 129-168. Teitelbaum, H. & Englander, S. W. (1975) J . Mol. Biol. Y2, 55 - 92. Giacomoni, P. U., Delain, E. & Le Pecq, J. B. (1977) Eur. J . Biochem. 78,205-213. Burgess, R. R. (1976) in R N A Polymerase (Losik, R. & Chamberlin, M. J., eds) pp. 69-100, Cold Spring Harbor Laboratories, New York. Giacomoni, P. U., Delain, E. & Le Pecq, J. B. (1977) Eur. J . Biochem. 78,215-220. Lin, S. Y. & Riggs, A. D. (1972) J . Mol. Biol. 72, 671 -690. Seeburg, P. H. & Schaller, H. (1975)J. Mol. Biol. 92,261-277.

16. de Haseth, P. L., Lohman, T. M., Burgess, R. R. & Record, M. T. (1978) Biochemistry, 17, 1612- 1622. 17. Mc Ghee, J. D. & Von Hippel, P. H. (1974) J . Mol. B i d . 86, 469 - 489. 18. Richter, P. H. & Eigen, M. (1974) Biophys. Chem. 2, 255-263. 19. Lohman, T. M., de Haseth, P. L. & Record, M. T. (1978) Biophys. Chem. 8,281 -294. 20. LeTalaer, J. Y. & Jeanteur, Ph. (1973) Proc. Nut1 Acad. Sci. U.S.A. 70,2911-2915. 21. von Gabain, A. & Bujard, H. (1977) Mol. Gen. Genet. 157, 301 -31 1. 22. Stueber, D., Delius, H. & Bujard, H. (1978) Mol. Gen. Genet. 166, 141 - 149. 23. Hinkle, D. C. s( Chamberlin, M. J. (1972) J . Mol. Bid. 70, 187- 195. 24. Einstein, A. (1905) Ann. Phys. 17, 549-560.

P. Giacomoni, Laboratoire de Pharmacologie Molkculaire du C.N.R.S., Institut Gustdve-Roussy, 16 bis Avenue Paul-Vaillant-Couturier, F-94800 Villejuif, France